{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0001","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e0839df24a10a918ba357031f0846e7fe02f8835f84c217ce18800fa5450d720","family_id":"qf-mod31-fielddeep-characteristic","seed":202610316001,"semantic_template":"qf-mod31-fielddeep-characteristic/{\"p\":2}"},"fields":{"title":"Characteristic Is the Additive Order of One","stem":"Use the complete prime field Z_2, with prime p=2, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Give (p,successive_sums_of_one,characteristic,additive_order_of_one). List k*1 for k=0 through p in that order.","answer":{"kind":"expression","value":"(2, [0, 1, 0], 2, additive_order_of_one)","explanation":"The residues are 0,1,...,p−1,0. The first positive number of copies of one giving zero is p, not a multiplicative element order. The exact result, retaining the requested order, is (2, [0, 1, 0], 2, additive_order_of_one)."},"skills":["Characteristic Is the Additive Order of One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0002","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e4eb300a861f33b9c568e773e247fb7a780018a15f56d2f4b377d2b9f1d91ea1","family_id":"qf-mod31-fielddeep-characteristic","seed":202610316002,"semantic_template":"qf-mod31-fielddeep-characteristic/{\"p\":3}"},"fields":{"title":"Characteristic Is the Additive Order of One","stem":"Use the complete prime field Z_3, with prime p=3, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Give (p,successive_sums_of_one,characteristic,additive_order_of_one). List k*1 for k=0 through p in that order.","answer":{"kind":"expression","value":"(3, [0, 1, 2, 0], 3, additive_order_of_one)","explanation":"The residues are 0,1,...,p−1,0. The first positive number of copies of one giving zero is p, not a multiplicative element order. The exact result, retaining the requested order, is (3, [0, 1, 2, 0], 3, additive_order_of_one)."},"skills":["Characteristic Is the Additive Order of One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0003","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b2f29b8008d5d366ead741443fa0fa689d6fa0daf3da0e80f33a8e987ee2e83b","family_id":"qf-mod31-fielddeep-characteristic","seed":202610316003,"semantic_template":"qf-mod31-fielddeep-characteristic/{\"p\":5}"},"fields":{"title":"Characteristic Is the Additive Order of One","stem":"Use the complete prime field Z_5, with prime p=5, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Give (p,successive_sums_of_one,characteristic,additive_order_of_one). List k*1 for k=0 through p in that order.","answer":{"kind":"expression","value":"(5, [0, 1, 2, 3, 4, 0], 5, additive_order_of_one)","explanation":"The residues are 0,1,...,p−1,0. The first positive number of copies of one giving zero is p, not a multiplicative element order. The exact result, retaining the requested order, is (5, [0, 1, 2, 3, 4, 0], 5, additive_order_of_one)."},"skills":["Characteristic Is the Additive Order of One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0004","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"df4973739d46cc78e7a4180c4fbf5f3a04fe5f5e706f65f242cdfa06d8d1796f","family_id":"qf-mod31-fielddeep-characteristic","seed":202610316004,"semantic_template":"qf-mod31-fielddeep-characteristic/{\"p\":7}"},"fields":{"title":"Characteristic Is the Additive Order of One","stem":"Use the complete prime field Z_7, with prime p=7, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Give (p,successive_sums_of_one,characteristic,additive_order_of_one). List k*1 for k=0 through p in that order.","answer":{"kind":"expression","value":"(7, [0, 1, 2, 3, 4, 5, 6, 0], 7, additive_order_of_one)","explanation":"The residues are 0,1,...,p−1,0. The first positive number of copies of one giving zero is p, not a multiplicative element order. The exact result, retaining the requested order, is (7, [0, 1, 2, 3, 4, 5, 6, 0], 7, additive_order_of_one)."},"skills":["Characteristic Is the Additive Order of One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0005","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a6a8ec3d0a44664235a70cc6eaaebf77c1fc45fc5e0f2c0d660f45c4838b415","family_id":"qf-mod31-fielddeep-characteristic","seed":202610316005,"semantic_template":"qf-mod31-fielddeep-characteristic/{\"p\":11}"},"fields":{"title":"Characteristic Is the Additive Order of One","stem":"Use the complete prime field Z_11, with prime p=11, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Give (p,successive_sums_of_one,characteristic,additive_order_of_one). List k*1 for k=0 through p in that order.","answer":{"kind":"expression","value":"(11, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0], 11, additive_order_of_one)","explanation":"The residues are 0,1,...,p−1,0. The first positive number of copies of one giving zero is p, not a multiplicative element order. The exact result, retaining the requested order, is (11, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0], 11, additive_order_of_one)."},"skills":["Characteristic Is the Additive Order of One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0006","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d2cac1cb67fb1773a126a28955cb01894d6df60852a80e7712b72f92a90f554d","family_id":"qf-mod31-fielddeep-characteristic","seed":202610316006,"semantic_template":"qf-mod31-fielddeep-characteristic/{\"p\":13}"},"fields":{"title":"Characteristic Is the Additive Order of One","stem":"Use the complete prime field Z_13, with prime p=13, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Give (p,successive_sums_of_one,characteristic,additive_order_of_one). List k*1 for k=0 through p in that order.","answer":{"kind":"expression","value":"(13, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0], 13, additive_order_of_one)","explanation":"The residues are 0,1,...,p−1,0. The first positive number of copies of one giving zero is p, not a multiplicative element order. The exact result, retaining the requested order, is (13, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0], 13, additive_order_of_one)."},"skills":["Characteristic Is the Additive Order of One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0007","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fe5704576fc2846fb980e33b328dc75995d0b45232a680f0c1d8b09b71b21d64","family_id":"qf-mod31-fielddeep-characteristic","seed":202610316007,"semantic_template":"qf-mod31-fielddeep-characteristic/{\"p\":17}"},"fields":{"title":"Characteristic Is the Additive Order of One","stem":"Use the complete prime field Z_17, with prime p=17, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Give (p,successive_sums_of_one,characteristic,additive_order_of_one). List k*1 for k=0 through p in that order.","answer":{"kind":"expression","value":"(17, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0], 17, additive_order_of_one)","explanation":"The residues are 0,1,...,p−1,0. The first positive number of copies of one giving zero is p, not a multiplicative element order. The exact result, retaining the requested order, is (17, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0], 17, additive_order_of_one)."},"skills":["Characteristic Is the Additive Order of One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0008","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e4670dbee2cc7b0ca4d3ff414dcc1d492d1992ad1db468a6f2157be01191fad8","family_id":"qf-mod31-fielddeep-characteristic","seed":202610316008,"semantic_template":"qf-mod31-fielddeep-characteristic/{\"p\":19}"},"fields":{"title":"Characteristic Is the Additive Order of One","stem":"Use the complete prime field Z_19, with prime p=19, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Give (p,successive_sums_of_one,characteristic,additive_order_of_one). List k*1 for k=0 through p in that order.","answer":{"kind":"expression","value":"(19, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0], 19, additive_order_of_one)","explanation":"The residues are 0,1,...,p−1,0. The first positive number of copies of one giving zero is p, not a multiplicative element order. The exact result, retaining the requested order, is (19, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0], 19, additive_order_of_one)."},"skills":["Characteristic Is the Additive Order of One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0009","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cb0212bbe218a09925f69e12e3f8b3ff96e1df7639b05e049f8f3baca7f3258e","family_id":"qf-mod31-fielddeep-characteristic","seed":202610316009,"semantic_template":"qf-mod31-fielddeep-characteristic/{\"p\":23}"},"fields":{"title":"Characteristic Is the Additive Order of One","stem":"Use the complete prime field Z_23, with prime p=23, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Give (p,successive_sums_of_one,characteristic,additive_order_of_one). List k*1 for k=0 through p in that order.","answer":{"kind":"expression","value":"(23, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0], 23, additive_order_of_one)","explanation":"The residues are 0,1,...,p−1,0. The first positive number of copies of one giving zero is p, not a multiplicative element order. The exact result, retaining the requested order, is (23, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0], 23, additive_order_of_one)."},"skills":["Characteristic Is the Additive Order of One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0010","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"75ea169c9648111adff335094cd5460e040bbdaaea26aa03d88bfc67400244f4","family_id":"qf-mod31-fielddeep-characteristic","seed":202610316010,"semantic_template":"qf-mod31-fielddeep-characteristic/{\"p\":29}"},"fields":{"title":"Characteristic Is the Additive Order of One","stem":"Use the complete prime field Z_29, with prime p=29, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Give (p,successive_sums_of_one,characteristic,additive_order_of_one). List k*1 for k=0 through p in that order.","answer":{"kind":"expression","value":"(29, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 0], 29, additive_order_of_one)","explanation":"The residues are 0,1,...,p−1,0. The first positive number of copies of one giving zero is p, not a multiplicative element order. The exact result, retaining the requested order, is (29, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 0], 29, additive_order_of_one)."},"skills":["Characteristic Is the Additive Order of One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0011","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b24bafa1308b05deaf0cbf19da3cb92e3f305c92f1dc42af8ed64024883ec6c0","family_id":"qf-mod31-fielddeep-characteristic","seed":202610316011,"semantic_template":"qf-mod31-fielddeep-characteristic/{\"p\":31}"},"fields":{"title":"Characteristic Is the Additive Order of One","stem":"Use the complete prime field Z_31, with prime p=31, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Give (p,successive_sums_of_one,characteristic,additive_order_of_one). List k*1 for k=0 through p in that order.","answer":{"kind":"expression","value":"(31, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 0], 31, additive_order_of_one)","explanation":"The residues are 0,1,...,p−1,0. The first positive number of copies of one giving zero is p, not a multiplicative element order. The exact result, retaining the requested order, is (31, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 0], 31, additive_order_of_one)."},"skills":["Characteristic Is the Additive Order of One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0012","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"72c2c6a41f3fc9d84c48006c3d115c5b67ff3d3aa33e0c916093d69272665895","family_id":"qf-mod31-fielddeep-characteristic","seed":202610316012,"semantic_template":"qf-mod31-fielddeep-characteristic/{\"p\":37}"},"fields":{"title":"Characteristic Is the Additive Order of One","stem":"Use the complete prime field Z_37, with prime p=37, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Give (p,successive_sums_of_one,characteristic,additive_order_of_one). List k*1 for k=0 through p in that order.","answer":{"kind":"expression","value":"(37, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 0], 37, additive_order_of_one)","explanation":"The residues are 0,1,...,p−1,0. The first positive number of copies of one giving zero is p, not a multiplicative element order. The exact result, retaining the requested order, is (37, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 0], 37, additive_order_of_one)."},"skills":["Characteristic Is the Additive Order of One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0013","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"678f7a009e48ee589296467714e7208e1e58c695d0b6075b76e493cc4243ade3","family_id":"qf-mod31-fielddeep-characteristic","seed":202610316013,"semantic_template":"qf-mod31-fielddeep-characteristic/{\"p\":41}"},"fields":{"title":"Characteristic Is the Additive Order of One","stem":"Use the complete prime field Z_41, with prime p=41, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Give (p,successive_sums_of_one,characteristic,additive_order_of_one). List k*1 for k=0 through p in that order.","answer":{"kind":"expression","value":"(41, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 0], 41, additive_order_of_one)","explanation":"The residues are 0,1,...,p−1,0. The first positive number of copies of one giving zero is p, not a multiplicative element order. The exact result, retaining the requested order, is (41, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 0], 41, additive_order_of_one)."},"skills":["Characteristic Is the Additive Order of One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0014","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c00c398a871c9368a0ccfb143173b1072af4b734e2a180cfb83e7a8377e64701","family_id":"qf-mod31-fielddeep-characteristic","seed":202610316014,"semantic_template":"qf-mod31-fielddeep-characteristic/{\"p\":43}"},"fields":{"title":"Characteristic Is the Additive Order of One","stem":"Use the complete prime field Z_43, with prime p=43, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Give (p,successive_sums_of_one,characteristic,additive_order_of_one). List k*1 for k=0 through p in that order.","answer":{"kind":"expression","value":"(43, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 0], 43, additive_order_of_one)","explanation":"The residues are 0,1,...,p−1,0. The first positive number of copies of one giving zero is p, not a multiplicative element order. The exact result, retaining the requested order, is (43, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 0], 43, additive_order_of_one)."},"skills":["Characteristic Is the Additive Order of One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0015","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0fa046d90c235547a7d609b9b2103414f57b5019e2bbc8da3f61ae2664b458e7","family_id":"qf-mod31-fielddeep-characteristic","seed":202610316015,"semantic_template":"qf-mod31-fielddeep-characteristic/{\"p\":47}"},"fields":{"title":"Characteristic Is the Additive Order of One","stem":"Use the complete prime field Z_47, with prime p=47, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Give (p,successive_sums_of_one,characteristic,additive_order_of_one). List k*1 for k=0 through p in that order.","answer":{"kind":"expression","value":"(47, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 0], 47, additive_order_of_one)","explanation":"The residues are 0,1,...,p−1,0. The first positive number of copies of one giving zero is p, not a multiplicative element order. The exact result, retaining the requested order, is (47, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 0], 47, additive_order_of_one)."},"skills":["Characteristic Is the Additive Order of One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0016","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d156862c25c5f635e7df7a0271d19ac45aa17740e3f6b9d6ae45e5c613424ce6","family_id":"qf-mod31-fielddeep-characteristic","seed":202610316016,"semantic_template":"qf-mod31-fielddeep-characteristic/{\"p\":53}"},"fields":{"title":"Characteristic Is the Additive Order of One","stem":"Use the complete prime field Z_53, with prime p=53, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Give (p,successive_sums_of_one,characteristic,additive_order_of_one). List k*1 for k=0 through p in that order.","answer":{"kind":"expression","value":"(53, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 0], 53, additive_order_of_one)","explanation":"The residues are 0,1,...,p−1,0. The first positive number of copies of one giving zero is p, not a multiplicative element order. The exact result, retaining the requested order, is (53, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 0], 53, additive_order_of_one)."},"skills":["Characteristic Is the Additive Order of One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0017","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"130c8a4056f51320fdc958a833ad2f10377ef716566b1b3ec6c60726129c2754","family_id":"qf-mod31-fielddeep-characteristic","seed":202610316017,"semantic_template":"qf-mod31-fielddeep-characteristic/{\"p\":59}"},"fields":{"title":"Characteristic Is the Additive Order of One","stem":"Use the complete prime field Z_59, with prime p=59, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Give (p,successive_sums_of_one,characteristic,additive_order_of_one). List k*1 for k=0 through p in that order.","answer":{"kind":"expression","value":"(59, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 0], 59, additive_order_of_one)","explanation":"The residues are 0,1,...,p−1,0. The first positive number of copies of one giving zero is p, not a multiplicative element order. The exact result, retaining the requested order, is (59, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 0], 59, additive_order_of_one)."},"skills":["Characteristic Is the Additive Order of One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0018","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"46608671479cc77f580707706460a7cdf787d5159f709d2b7590009624ddb261","family_id":"qf-mod31-fielddeep-characteristic","seed":202610316018,"semantic_template":"qf-mod31-fielddeep-characteristic/{\"p\":61}"},"fields":{"title":"Characteristic Is the Additive Order of One","stem":"Use the complete prime field Z_61, with prime p=61, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Give (p,successive_sums_of_one,characteristic,additive_order_of_one). List k*1 for k=0 through p in that order.","answer":{"kind":"expression","value":"(61, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 0], 61, additive_order_of_one)","explanation":"The residues are 0,1,...,p−1,0. The first positive number of copies of one giving zero is p, not a multiplicative element order. The exact result, retaining the requested order, is (61, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 0], 61, additive_order_of_one)."},"skills":["Characteristic Is the Additive Order of One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0019","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf8d30b896ee2677bbd0e3f3d6ef71fa07bf631d0fcc2c13f6824027130ce509","family_id":"qf-mod31-fielddeep-characteristic","seed":202610316019,"semantic_template":"qf-mod31-fielddeep-characteristic/{\"p\":67}"},"fields":{"title":"Characteristic Is the Additive Order of One","stem":"Use the complete prime field Z_67, with prime p=67, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Give (p,successive_sums_of_one,characteristic,additive_order_of_one). List k*1 for k=0 through p in that order.","answer":{"kind":"expression","value":"(67, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 0], 67, additive_order_of_one)","explanation":"The residues are 0,1,...,p−1,0. The first positive number of copies of one giving zero is p, not a multiplicative element order. The exact result, retaining the requested order, is (67, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 0], 67, additive_order_of_one)."},"skills":["Characteristic Is the Additive Order of One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0020","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ca1aabec37de406259d58d263bef5dbc918acd6ff18ae7787b5729a4f81df887","family_id":"qf-mod31-fielddeep-characteristic","seed":202610316020,"semantic_template":"qf-mod31-fielddeep-characteristic/{\"p\":71}"},"fields":{"title":"Characteristic Is the Additive Order of One","stem":"Use the complete prime field Z_71, with prime p=71, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Give (p,successive_sums_of_one,characteristic,additive_order_of_one). List k*1 for k=0 through p in that order.","answer":{"kind":"expression","value":"(71, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 0], 71, additive_order_of_one)","explanation":"The residues are 0,1,...,p−1,0. The first positive number of copies of one giving zero is p, not a multiplicative element order. The exact result, retaining the requested order, is (71, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 0], 71, additive_order_of_one)."},"skills":["Characteristic Is the Additive Order of One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0021","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"be31defa1b64c0717c09b1b927219bf5ef532d8cdebbdf81fbcd4a322fcd8892","family_id":"qf-mod31-fielddeep-characteristic","seed":202610316021,"semantic_template":"qf-mod31-fielddeep-characteristic/{\"p\":73}"},"fields":{"title":"Characteristic Is the Additive Order of One","stem":"Use the complete prime field Z_73, with prime p=73, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Give (p,successive_sums_of_one,characteristic,additive_order_of_one). List k*1 for k=0 through p in that order.","answer":{"kind":"expression","value":"(73, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 0], 73, additive_order_of_one)","explanation":"The residues are 0,1,...,p−1,0. The first positive number of copies of one giving zero is p, not a multiplicative element order. The exact result, retaining the requested order, is (73, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 0], 73, additive_order_of_one)."},"skills":["Characteristic Is the Additive Order of One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0022","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f2377c51c8593adb01f2bdc0c633b122babf7564ce0930b433d34919196bde75","family_id":"qf-mod31-fielddeep-characteristic","seed":202610316022,"semantic_template":"qf-mod31-fielddeep-characteristic/{\"p\":79}"},"fields":{"title":"Characteristic Is the Additive Order of One","stem":"Use the complete prime field Z_79, with prime p=79, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Give (p,successive_sums_of_one,characteristic,additive_order_of_one). List k*1 for k=0 through p in that order.","answer":{"kind":"expression","value":"(79, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 0], 79, additive_order_of_one)","explanation":"The residues are 0,1,...,p−1,0. The first positive number of copies of one giving zero is p, not a multiplicative element order. The exact result, retaining the requested order, is (79, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 0], 79, additive_order_of_one)."},"skills":["Characteristic Is the Additive Order of One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0023","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ae2c80baff7c7873ccac1e7c0a073ccff73b8391d77ec01b500a8700b0262b50","family_id":"qf-mod31-fielddeep-characteristic","seed":202610316023,"semantic_template":"qf-mod31-fielddeep-characteristic/{\"p\":83}"},"fields":{"title":"Characteristic Is the Additive Order of One","stem":"Use the complete prime field Z_83, with prime p=83, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Give (p,successive_sums_of_one,characteristic,additive_order_of_one). List k*1 for k=0 through p in that order.","answer":{"kind":"expression","value":"(83, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 0], 83, additive_order_of_one)","explanation":"The residues are 0,1,...,p−1,0. The first positive number of copies of one giving zero is p, not a multiplicative element order. The exact result, retaining the requested order, is (83, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 0], 83, additive_order_of_one)."},"skills":["Characteristic Is the Additive Order of One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0024","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7bc1e5199558c8e84e7de66ba58afd878b0dfedefbbd834396f5aa29b07c317d","family_id":"qf-mod31-fielddeep-characteristic","seed":202610316024,"semantic_template":"qf-mod31-fielddeep-characteristic/{\"p\":89}"},"fields":{"title":"Characteristic Is the Additive Order of One","stem":"Use the complete prime field Z_89, with prime p=89, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Give (p,successive_sums_of_one,characteristic,additive_order_of_one). List k*1 for k=0 through p in that order.","answer":{"kind":"expression","value":"(89, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 0], 89, additive_order_of_one)","explanation":"The residues are 0,1,...,p−1,0. The first positive number of copies of one giving zero is p, not a multiplicative element order. The exact result, retaining the requested order, is (89, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 0], 89, additive_order_of_one)."},"skills":["Characteristic Is the Additive Order of One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0025","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e6c60559d55655a383544ba7bc217b9bb46d5083898556b89fb6aee71118a47","family_id":"qf-mod31-fielddeep-characteristic","seed":202610316025,"semantic_template":"qf-mod31-fielddeep-characteristic/{\"p\":97}"},"fields":{"title":"Characteristic Is the Additive Order of One","stem":"Use the complete prime field Z_97, with prime p=97, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Give (p,successive_sums_of_one,characteristic,additive_order_of_one). List k*1 for k=0 through p in that order.","answer":{"kind":"expression","value":"(97, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 0], 97, additive_order_of_one)","explanation":"The residues are 0,1,...,p−1,0. The first positive number of copies of one giving zero is p, not a multiplicative element order. The exact result, retaining the requested order, is (97, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 0], 97, additive_order_of_one)."},"skills":["Characteristic Is the Additive Order of One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0026","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"148a39dd8be9f78ab74cc5aeab6c2e111f2e799fc825391c0debef4cd67bb25a","family_id":"qf-mod31-fielddeep-power-cycle","seed":202610316026,"semantic_template":"qf-mod31-fielddeep-power-cycle/{\"element\":1,\"p\":2}"},"fields":{"title":"The Nonzero Multiplicative Group Gives a p−1 Power Identity","stem":"Use the complete prime field Z_2, with prime p=2, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero element=1. Give (element,powers,last,is_one?,nonzero_multiplicative_group_cycle), listing exponents 1 through p−1.","answer":{"kind":"expression","value":"(1, [1], 1, Yes, nonzero_multiplicative_group_cycle)","explanation":"Compute powers modulo p in exponent order. The final power equals one, but the element may have a smaller multiplicative order; p−1 is not asserted to be its exact order or a primitive-generator certificate. The exact result, retaining the requested order, is (1, [1], 1, Yes, nonzero_multiplicative_group_cycle)."},"skills":["The Nonzero Multiplicative Group Gives a p−1 Power Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0027","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a9f913f2688534d09d84416aed05b5b69458e3d45ed3a096e5671a8b8634bb90","family_id":"qf-mod31-fielddeep-power-cycle","seed":202610316027,"semantic_template":"qf-mod31-fielddeep-power-cycle/{\"element\":2,\"p\":3}"},"fields":{"title":"The Nonzero Multiplicative Group Gives a p−1 Power Identity","stem":"Use the complete prime field Z_3, with prime p=3, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero element=2. Give (element,powers,last,is_one?,nonzero_multiplicative_group_cycle), listing exponents 1 through p−1.","answer":{"kind":"expression","value":"(2, [2, 1], 1, Yes, nonzero_multiplicative_group_cycle)","explanation":"Compute powers modulo p in exponent order. The final power equals one, but the element may have a smaller multiplicative order; p−1 is not asserted to be its exact order or a primitive-generator certificate. The exact result, retaining the requested order, is (2, [2, 1], 1, Yes, nonzero_multiplicative_group_cycle)."},"skills":["The Nonzero Multiplicative Group Gives a p−1 Power Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0028","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3ed2387eedc3219e7fd5eb7be02e463a6e397c198a5e49509b33203e1c38ff7b","family_id":"qf-mod31-fielddeep-power-cycle","seed":202610316028,"semantic_template":"qf-mod31-fielddeep-power-cycle/{\"element\":3,\"p\":5}"},"fields":{"title":"The Nonzero Multiplicative Group Gives a p−1 Power Identity","stem":"Use the complete prime field Z_5, with prime p=5, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero element=3. Give (element,powers,last,is_one?,nonzero_multiplicative_group_cycle), listing exponents 1 through p−1.","answer":{"kind":"expression","value":"(3, [3, 4, 2, 1], 1, Yes, nonzero_multiplicative_group_cycle)","explanation":"Compute powers modulo p in exponent order. The final power equals one, but the element may have a smaller multiplicative order; p−1 is not asserted to be its exact order or a primitive-generator certificate. The exact result, retaining the requested order, is (3, [3, 4, 2, 1], 1, Yes, nonzero_multiplicative_group_cycle)."},"skills":["The Nonzero Multiplicative Group Gives a p−1 Power Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0029","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"37c9936029aa8186580a81107525fdfb26c90015b992fa0355fd473c03d9cd08","family_id":"qf-mod31-fielddeep-power-cycle","seed":202610316029,"semantic_template":"qf-mod31-fielddeep-power-cycle/{\"element\":4,\"p\":7}"},"fields":{"title":"The Nonzero Multiplicative Group Gives a p−1 Power Identity","stem":"Use the complete prime field Z_7, with prime p=7, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero element=4. Give (element,powers,last,is_one?,nonzero_multiplicative_group_cycle), listing exponents 1 through p−1.","answer":{"kind":"expression","value":"(4, [4, 2, 1, 4, 2, 1], 1, Yes, nonzero_multiplicative_group_cycle)","explanation":"Compute powers modulo p in exponent order. The final power equals one, but the element may have a smaller multiplicative order; p−1 is not asserted to be its exact order or a primitive-generator certificate. The exact result, retaining the requested order, is (4, [4, 2, 1, 4, 2, 1], 1, Yes, nonzero_multiplicative_group_cycle)."},"skills":["The Nonzero Multiplicative Group Gives a p−1 Power Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0030","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d17cc6313b23b6f16dd470a5e9075699501d964a30adee601bfd7ff5494d18b2","family_id":"qf-mod31-fielddeep-power-cycle","seed":202610316030,"semantic_template":"qf-mod31-fielddeep-power-cycle/{\"element\":5,\"p\":11}"},"fields":{"title":"The Nonzero Multiplicative Group Gives a p−1 Power Identity","stem":"Use the complete prime field Z_11, with prime p=11, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero element=5. Give (element,powers,last,is_one?,nonzero_multiplicative_group_cycle), listing exponents 1 through p−1.","answer":{"kind":"expression","value":"(5, [5, 3, 4, 9, 1, 5, 3, 4, 9, 1], 1, Yes, nonzero_multiplicative_group_cycle)","explanation":"Compute powers modulo p in exponent order. The final power equals one, but the element may have a smaller multiplicative order; p−1 is not asserted to be its exact order or a primitive-generator certificate. The exact result, retaining the requested order, is (5, [5, 3, 4, 9, 1, 5, 3, 4, 9, 1], 1, Yes, nonzero_multiplicative_group_cycle)."},"skills":["The Nonzero Multiplicative Group Gives a p−1 Power Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0031","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c90861c75cf128e0dcb7d979d43d1c719543a0e9fbb7cd13bc12aff09051b07b","family_id":"qf-mod31-fielddeep-power-cycle","seed":202610316031,"semantic_template":"qf-mod31-fielddeep-power-cycle/{\"element\":6,\"p\":13}"},"fields":{"title":"The Nonzero Multiplicative Group Gives a p−1 Power Identity","stem":"Use the complete prime field Z_13, with prime p=13, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero element=6. Give (element,powers,last,is_one?,nonzero_multiplicative_group_cycle), listing exponents 1 through p−1.","answer":{"kind":"expression","value":"(6, [6, 10, 8, 9, 2, 12, 7, 3, 5, 4, 11, 1], 1, Yes, nonzero_multiplicative_group_cycle)","explanation":"Compute powers modulo p in exponent order. The final power equals one, but the element may have a smaller multiplicative order; p−1 is not asserted to be its exact order or a primitive-generator certificate. The exact result, retaining the requested order, is (6, [6, 10, 8, 9, 2, 12, 7, 3, 5, 4, 11, 1], 1, Yes, nonzero_multiplicative_group_cycle)."},"skills":["The Nonzero Multiplicative Group Gives a p−1 Power Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0032","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"092b4b591e2f9cf671aeb3362ab874c40bdb7fdaa646dcc5216687778b21714a","family_id":"qf-mod31-fielddeep-power-cycle","seed":202610316032,"semantic_template":"qf-mod31-fielddeep-power-cycle/{\"element\":7,\"p\":17}"},"fields":{"title":"The Nonzero Multiplicative Group Gives a p−1 Power Identity","stem":"Use the complete prime field Z_17, with prime p=17, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero element=7. Give (element,powers,last,is_one?,nonzero_multiplicative_group_cycle), listing exponents 1 through p−1.","answer":{"kind":"expression","value":"(7, [7, 15, 3, 4, 11, 9, 12, 16, 10, 2, 14, 13, 6, 8, 5, 1], 1, Yes, nonzero_multiplicative_group_cycle)","explanation":"Compute powers modulo p in exponent order. The final power equals one, but the element may have a smaller multiplicative order; p−1 is not asserted to be its exact order or a primitive-generator certificate. The exact result, retaining the requested order, is (7, [7, 15, 3, 4, 11, 9, 12, 16, 10, 2, 14, 13, 6, 8, 5, 1], 1, Yes, nonzero_multiplicative_group_cycle)."},"skills":["The Nonzero Multiplicative Group Gives a p−1 Power Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0033","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5b67e5310e548d261a9cf6f58ce0bac8e84d4c7d10db38a9848b8e0d2c1da345","family_id":"qf-mod31-fielddeep-power-cycle","seed":202610316033,"semantic_template":"qf-mod31-fielddeep-power-cycle/{\"element\":8,\"p\":19}"},"fields":{"title":"The Nonzero Multiplicative Group Gives a p−1 Power Identity","stem":"Use the complete prime field Z_19, with prime p=19, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero element=8. Give (element,powers,last,is_one?,nonzero_multiplicative_group_cycle), listing exponents 1 through p−1.","answer":{"kind":"expression","value":"(8, [8, 7, 18, 11, 12, 1, 8, 7, 18, 11, 12, 1, 8, 7, 18, 11, 12, 1], 1, Yes, nonzero_multiplicative_group_cycle)","explanation":"Compute powers modulo p in exponent order. The final power equals one, but the element may have a smaller multiplicative order; p−1 is not asserted to be its exact order or a primitive-generator certificate. The exact result, retaining the requested order, is (8, [8, 7, 18, 11, 12, 1, 8, 7, 18, 11, 12, 1, 8, 7, 18, 11, 12, 1], 1, Yes, nonzero_multiplicative_group_cycle)."},"skills":["The Nonzero Multiplicative Group Gives a p−1 Power Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0034","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"68aa95f9c6513e2dc37b48fcf73bba3a47d4072dbf469da0bab331e87d0ba627","family_id":"qf-mod31-fielddeep-power-cycle","seed":202610316034,"semantic_template":"qf-mod31-fielddeep-power-cycle/{\"element\":9,\"p\":23}"},"fields":{"title":"The Nonzero Multiplicative Group Gives a p−1 Power Identity","stem":"Use the complete prime field Z_23, with prime p=23, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero element=9. Give (element,powers,last,is_one?,nonzero_multiplicative_group_cycle), listing exponents 1 through p−1.","answer":{"kind":"expression","value":"(9, [9, 12, 16, 6, 8, 3, 4, 13, 2, 18, 1, 9, 12, 16, 6, 8, 3, 4, 13, 2, 18, 1], 1, Yes, nonzero_multiplicative_group_cycle)","explanation":"Compute powers modulo p in exponent order. The final power equals one, but the element may have a smaller multiplicative order; p−1 is not asserted to be its exact order or a primitive-generator certificate. The exact result, retaining the requested order, is (9, [9, 12, 16, 6, 8, 3, 4, 13, 2, 18, 1, 9, 12, 16, 6, 8, 3, 4, 13, 2, 18, 1], 1, Yes, nonzero_multiplicative_group_cycle)."},"skills":["The Nonzero Multiplicative Group Gives a p−1 Power Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0035","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"435f861f89141c7e180f4ef0334fcd02db2192560f4f297fc17db63b41f309eb","family_id":"qf-mod31-fielddeep-power-cycle","seed":202610316035,"semantic_template":"qf-mod31-fielddeep-power-cycle/{\"element\":10,\"p\":29}"},"fields":{"title":"The Nonzero Multiplicative Group Gives a p−1 Power Identity","stem":"Use the complete prime field Z_29, with prime p=29, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero element=10. Give (element,powers,last,is_one?,nonzero_multiplicative_group_cycle), listing exponents 1 through p−1.","answer":{"kind":"expression","value":"(10, [10, 13, 14, 24, 8, 22, 17, 25, 18, 6, 2, 20, 26, 28, 19, 16, 15, 5, 21, 7, 12, 4, 11, 23, 27, 9, 3, 1], 1, Yes, nonzero_multiplicative_group_cycle)","explanation":"Compute powers modulo p in exponent order. The final power equals one, but the element may have a smaller multiplicative order; p−1 is not asserted to be its exact order or a primitive-generator certificate. The exact result, retaining the requested order, is (10, [10, 13, 14, 24, 8, 22, 17, 25, 18, 6, 2, 20, 26, 28, 19, 16, 15, 5, 21, 7, 12, 4, 11, 23, 27, 9, 3, 1], 1, Yes, nonzero_multiplicative_group_cycle)."},"skills":["The Nonzero Multiplicative Group Gives a p−1 Power Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0036","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"22df5ba33e3ec71d37f501cc6c3a49eea07c354dfe47c4f4678b62514403c8d9","family_id":"qf-mod31-fielddeep-power-cycle","seed":202610316036,"semantic_template":"qf-mod31-fielddeep-power-cycle/{\"element\":11,\"p\":31}"},"fields":{"title":"The Nonzero Multiplicative Group Gives a p−1 Power Identity","stem":"Use the complete prime field Z_31, with prime p=31, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero element=11. Give (element,powers,last,is_one?,nonzero_multiplicative_group_cycle), listing exponents 1 through p−1.","answer":{"kind":"expression","value":"(11, [11, 28, 29, 9, 6, 4, 13, 19, 23, 5, 24, 16, 21, 14, 30, 20, 3, 2, 22, 25, 27, 18, 12, 8, 26, 7, 15, 10, 17, 1], 1, Yes, nonzero_multiplicative_group_cycle)","explanation":"Compute powers modulo p in exponent order. The final power equals one, but the element may have a smaller multiplicative order; p−1 is not asserted to be its exact order or a primitive-generator certificate. The exact result, retaining the requested order, is (11, [11, 28, 29, 9, 6, 4, 13, 19, 23, 5, 24, 16, 21, 14, 30, 20, 3, 2, 22, 25, 27, 18, 12, 8, 26, 7, 15, 10, 17, 1], 1, Yes, nonzero_multiplicative_group_cycle)."},"skills":["The Nonzero Multiplicative Group Gives a p−1 Power Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0037","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b46c83570440d45f12b527b2d739c181ff327e01f5b2c6f3de68de03b538704f","family_id":"qf-mod31-fielddeep-power-cycle","seed":202610316037,"semantic_template":"qf-mod31-fielddeep-power-cycle/{\"element\":12,\"p\":37}"},"fields":{"title":"The Nonzero Multiplicative Group Gives a p−1 Power Identity","stem":"Use the complete prime field Z_37, with prime p=37, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero element=12. Give (element,powers,last,is_one?,nonzero_multiplicative_group_cycle), listing exponents 1 through p−1.","answer":{"kind":"expression","value":"(12, [12, 33, 26, 16, 7, 10, 9, 34, 1, 12, 33, 26, 16, 7, 10, 9, 34, 1, 12, 33, 26, 16, 7, 10, 9, 34, 1, 12, 33, 26, 16, 7, 10, 9, 34, 1], 1, Yes, nonzero_multiplicative_group_cycle)","explanation":"Compute powers modulo p in exponent order. The final power equals one, but the element may have a smaller multiplicative order; p−1 is not asserted to be its exact order or a primitive-generator certificate. The exact result, retaining the requested order, is (12, [12, 33, 26, 16, 7, 10, 9, 34, 1, 12, 33, 26, 16, 7, 10, 9, 34, 1, 12, 33, 26, 16, 7, 10, 9, 34, 1, 12, 33, 26, 16, 7, 10, 9, 34, 1], 1, Yes, nonzero_multiplicative_group_cycle)."},"skills":["The Nonzero Multiplicative Group Gives a p−1 Power Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0038","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cedf00756071839c4c2d9c24d0138085c074a80634652f5c96478d7016d8c4a7","family_id":"qf-mod31-fielddeep-power-cycle","seed":202610316038,"semantic_template":"qf-mod31-fielddeep-power-cycle/{\"element\":13,\"p\":41}"},"fields":{"title":"The Nonzero Multiplicative Group Gives a p−1 Power Identity","stem":"Use the complete prime field Z_41, with prime p=41, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero element=13. Give (element,powers,last,is_one?,nonzero_multiplicative_group_cycle), listing exponents 1 through p−1.","answer":{"kind":"expression","value":"(13, [13, 5, 24, 25, 38, 2, 26, 10, 7, 9, 35, 4, 11, 20, 14, 18, 29, 8, 22, 40, 28, 36, 17, 16, 3, 39, 15, 31, 34, 32, 6, 37, 30, 21, 27, 23, 12, 33, 19, 1], 1, Yes, nonzero_multiplicative_group_cycle)","explanation":"Compute powers modulo p in exponent order. The final power equals one, but the element may have a smaller multiplicative order; p−1 is not asserted to be its exact order or a primitive-generator certificate. The exact result, retaining the requested order, is (13, [13, 5, 24, 25, 38, 2, 26, 10, 7, 9, 35, 4, 11, 20, 14, 18, 29, 8, 22, 40, 28, 36, 17, 16, 3, 39, 15, 31, 34, 32, 6, 37, 30, 21, 27, 23, 12, 33, 19, 1], 1, Yes, nonzero_multiplicative_group_cycle)."},"skills":["The Nonzero Multiplicative Group Gives a p−1 Power Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0039","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"800ecbad420d206661c9cdf9e97b4587e5e413d13e48d767b1753ab97534d27c","family_id":"qf-mod31-fielddeep-power-cycle","seed":202610316039,"semantic_template":"qf-mod31-fielddeep-power-cycle/{\"element\":14,\"p\":43}"},"fields":{"title":"The Nonzero Multiplicative Group Gives a p−1 Power Identity","stem":"Use the complete prime field Z_43, with prime p=43, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero element=14. Give (element,powers,last,is_one?,nonzero_multiplicative_group_cycle), listing exponents 1 through p−1.","answer":{"kind":"expression","value":"(14, [14, 24, 35, 17, 23, 21, 36, 31, 4, 13, 10, 11, 25, 6, 41, 15, 38, 16, 9, 40, 1, 14, 24, 35, 17, 23, 21, 36, 31, 4, 13, 10, 11, 25, 6, 41, 15, 38, 16, 9, 40, 1], 1, Yes, nonzero_multiplicative_group_cycle)","explanation":"Compute powers modulo p in exponent order. The final power equals one, but the element may have a smaller multiplicative order; p−1 is not asserted to be its exact order or a primitive-generator certificate. The exact result, retaining the requested order, is (14, [14, 24, 35, 17, 23, 21, 36, 31, 4, 13, 10, 11, 25, 6, 41, 15, 38, 16, 9, 40, 1, 14, 24, 35, 17, 23, 21, 36, 31, 4, 13, 10, 11, 25, 6, 41, 15, 38, 16, 9, 40, 1], 1, Yes, nonzero_multiplicative_group_cycle)."},"skills":["The Nonzero Multiplicative Group Gives a p−1 Power Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0040","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7c7a1eb2426746e94b61b42f749a53fa6098ded05ca0e6d80e7eb5818154d42c","family_id":"qf-mod31-fielddeep-power-cycle","seed":202610316040,"semantic_template":"qf-mod31-fielddeep-power-cycle/{\"element\":15,\"p\":47}"},"fields":{"title":"The Nonzero Multiplicative Group Gives a p−1 Power Identity","stem":"Use the complete prime field Z_47, with prime p=47, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero element=15. Give (element,powers,last,is_one?,nonzero_multiplicative_group_cycle), listing exponents 1 through p−1.","answer":{"kind":"expression","value":"(15, [15, 37, 38, 6, 43, 34, 40, 36, 23, 16, 5, 28, 44, 2, 30, 27, 29, 12, 39, 21, 33, 25, 46, 32, 10, 9, 41, 4, 13, 7, 11, 24, 31, 42, 19, 3, 45, 17, 20, 18, 35, 8, 26, 14, 22, 1], 1, Yes, nonzero_multiplicative_group_cycle)","explanation":"Compute powers modulo p in exponent order. The final power equals one, but the element may have a smaller multiplicative order; p−1 is not asserted to be its exact order or a primitive-generator certificate. The exact result, retaining the requested order, is (15, [15, 37, 38, 6, 43, 34, 40, 36, 23, 16, 5, 28, 44, 2, 30, 27, 29, 12, 39, 21, 33, 25, 46, 32, 10, 9, 41, 4, 13, 7, 11, 24, 31, 42, 19, 3, 45, 17, 20, 18, 35, 8, 26, 14, 22, 1], 1, Yes, nonzero_multiplicative_group_cycle)."},"skills":["The Nonzero Multiplicative Group Gives a p−1 Power Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0041","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8139615eea6f756df4c537528142b21837e059b840cf15c916a51f683684e3f0","family_id":"qf-mod31-fielddeep-power-cycle","seed":202610316041,"semantic_template":"qf-mod31-fielddeep-power-cycle/{\"element\":16,\"p\":53}"},"fields":{"title":"The Nonzero Multiplicative Group Gives a p−1 Power Identity","stem":"Use the complete prime field Z_53, with prime p=53, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero element=16. Give (element,powers,last,is_one?,nonzero_multiplicative_group_cycle), listing exponents 1 through p−1.","answer":{"kind":"expression","value":"(16, [16, 44, 15, 28, 24, 13, 49, 42, 36, 46, 47, 10, 1, 16, 44, 15, 28, 24, 13, 49, 42, 36, 46, 47, 10, 1, 16, 44, 15, 28, 24, 13, 49, 42, 36, 46, 47, 10, 1, 16, 44, 15, 28, 24, 13, 49, 42, 36, 46, 47, 10, 1], 1, Yes, nonzero_multiplicative_group_cycle)","explanation":"Compute powers modulo p in exponent order. The final power equals one, but the element may have a smaller multiplicative order; p−1 is not asserted to be its exact order or a primitive-generator certificate. The exact result, retaining the requested order, is (16, [16, 44, 15, 28, 24, 13, 49, 42, 36, 46, 47, 10, 1, 16, 44, 15, 28, 24, 13, 49, 42, 36, 46, 47, 10, 1, 16, 44, 15, 28, 24, 13, 49, 42, 36, 46, 47, 10, 1, 16, 44, 15, 28, 24, 13, 49, 42, 36, 46, 47, 10, 1], 1, Yes, nonzero_multiplicative_group_cycle)."},"skills":["The Nonzero Multiplicative Group Gives a p−1 Power Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0042","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3060c9a6ba3cff884cbf1834ec5e264ee964c454628265ee7eabffa18ada44f7","family_id":"qf-mod31-fielddeep-power-cycle","seed":202610316042,"semantic_template":"qf-mod31-fielddeep-power-cycle/{\"element\":17,\"p\":59}"},"fields":{"title":"The Nonzero Multiplicative Group Gives a p−1 Power Identity","stem":"Use the complete prime field Z_59, with prime p=59, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero element=17. Give (element,powers,last,is_one?,nonzero_multiplicative_group_cycle), listing exponents 1 through p−1.","answer":{"kind":"expression","value":"(17, [17, 53, 16, 36, 22, 20, 45, 57, 25, 12, 27, 46, 15, 19, 28, 4, 9, 35, 5, 26, 29, 21, 3, 51, 41, 48, 49, 7, 1, 17, 53, 16, 36, 22, 20, 45, 57, 25, 12, 27, 46, 15, 19, 28, 4, 9, 35, 5, 26, 29, 21, 3, 51, 41, 48, 49, 7, 1], 1, Yes, nonzero_multiplicative_group_cycle)","explanation":"Compute powers modulo p in exponent order. The final power equals one, but the element may have a smaller multiplicative order; p−1 is not asserted to be its exact order or a primitive-generator certificate. The exact result, retaining the requested order, is (17, [17, 53, 16, 36, 22, 20, 45, 57, 25, 12, 27, 46, 15, 19, 28, 4, 9, 35, 5, 26, 29, 21, 3, 51, 41, 48, 49, 7, 1, 17, 53, 16, 36, 22, 20, 45, 57, 25, 12, 27, 46, 15, 19, 28, 4, 9, 35, 5, 26, 29, 21, 3, 51, 41, 48, 49, 7, 1], 1, Yes, nonzero_multiplicative_group_cycle)."},"skills":["The Nonzero Multiplicative Group Gives a p−1 Power Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0043","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"04ecaf19ce55a9b2dacc463fc411b7967ede3676bcd42a80a2e3adf419480472","family_id":"qf-mod31-fielddeep-power-cycle","seed":202610316043,"semantic_template":"qf-mod31-fielddeep-power-cycle/{\"element\":18,\"p\":61}"},"fields":{"title":"The Nonzero Multiplicative Group Gives a p−1 Power Identity","stem":"Use the complete prime field Z_61, with prime p=61, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero element=18. Give (element,powers,last,is_one?,nonzero_multiplicative_group_cycle), listing exponents 1 through p−1.","answer":{"kind":"expression","value":"(18, [18, 19, 37, 56, 32, 27, 59, 25, 23, 48, 10, 58, 7, 4, 11, 15, 26, 41, 6, 47, 53, 39, 31, 9, 40, 49, 28, 16, 44, 60, 43, 42, 24, 5, 29, 34, 2, 36, 38, 13, 51, 3, 54, 57, 50, 46, 35, 20, 55, 14, 8, 22, 30, 52, 21, 12, 33, 45, 17, 1], 1, Yes, nonzero_multiplicative_group_cycle)","explanation":"Compute powers modulo p in exponent order. The final power equals one, but the element may have a smaller multiplicative order; p−1 is not asserted to be its exact order or a primitive-generator certificate. The exact result, retaining the requested order, is (18, [18, 19, 37, 56, 32, 27, 59, 25, 23, 48, 10, 58, 7, 4, 11, 15, 26, 41, 6, 47, 53, 39, 31, 9, 40, 49, 28, 16, 44, 60, 43, 42, 24, 5, 29, 34, 2, 36, 38, 13, 51, 3, 54, 57, 50, 46, 35, 20, 55, 14, 8, 22, 30, 52, 21, 12, 33, 45, 17, 1], 1, Yes, nonzero_multiplicative_group_cycle)."},"skills":["The Nonzero Multiplicative Group Gives a p−1 Power Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0044","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ae09b279346cecee3f2f753d14f0caf2b6815784f81db568d80ca34a40ced18f","family_id":"qf-mod31-fielddeep-power-cycle","seed":202610316044,"semantic_template":"qf-mod31-fielddeep-power-cycle/{\"element\":19,\"p\":67}"},"fields":{"title":"The Nonzero Multiplicative Group Gives a p−1 Power Identity","stem":"Use the complete prime field Z_67, with prime p=67, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero element=19. Give (element,powers,last,is_one?,nonzero_multiplicative_group_cycle), listing exponents 1 through p−1.","answer":{"kind":"expression","value":"(19, [19, 26, 25, 6, 47, 22, 16, 36, 14, 65, 29, 15, 17, 55, 40, 23, 35, 62, 39, 4, 9, 37, 33, 24, 54, 21, 64, 10, 56, 59, 49, 60, 1, 19, 26, 25, 6, 47, 22, 16, 36, 14, 65, 29, 15, 17, 55, 40, 23, 35, 62, 39, 4, 9, 37, 33, 24, 54, 21, 64, 10, 56, 59, 49, 60, 1], 1, Yes, nonzero_multiplicative_group_cycle)","explanation":"Compute powers modulo p in exponent order. The final power equals one, but the element may have a smaller multiplicative order; p−1 is not asserted to be its exact order or a primitive-generator certificate. The exact result, retaining the requested order, is (19, [19, 26, 25, 6, 47, 22, 16, 36, 14, 65, 29, 15, 17, 55, 40, 23, 35, 62, 39, 4, 9, 37, 33, 24, 54, 21, 64, 10, 56, 59, 49, 60, 1, 19, 26, 25, 6, 47, 22, 16, 36, 14, 65, 29, 15, 17, 55, 40, 23, 35, 62, 39, 4, 9, 37, 33, 24, 54, 21, 64, 10, 56, 59, 49, 60, 1], 1, Yes, nonzero_multiplicative_group_cycle)."},"skills":["The Nonzero Multiplicative Group Gives a p−1 Power Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0045","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b38eedd5c9569a695e6f87443d9b668a95e9b989a7dc0e593f8c0485a1c24d88","family_id":"qf-mod31-fielddeep-power-cycle","seed":202610316045,"semantic_template":"qf-mod31-fielddeep-power-cycle/{\"element\":20,\"p\":71}"},"fields":{"title":"The Nonzero Multiplicative Group Gives a p−1 Power Identity","stem":"Use the complete prime field Z_71, with prime p=71, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero element=20. Give (element,powers,last,is_one?,nonzero_multiplicative_group_cycle), listing exponents 1 through p−1.","answer":{"kind":"expression","value":"(20, [20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1], 1, Yes, nonzero_multiplicative_group_cycle)","explanation":"Compute powers modulo p in exponent order. The final power equals one, but the element may have a smaller multiplicative order; p−1 is not asserted to be its exact order or a primitive-generator certificate. The exact result, retaining the requested order, is (20, [20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1, 20, 45, 48, 37, 30, 32, 1], 1, Yes, nonzero_multiplicative_group_cycle)."},"skills":["The Nonzero Multiplicative Group Gives a p−1 Power Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0046","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7944a5b22c7f05c14a7eac1ba5148770ca63bf2282062a75032a0f5eef83957f","family_id":"qf-mod31-fielddeep-power-cycle","seed":202610316046,"semantic_template":"qf-mod31-fielddeep-power-cycle/{\"element\":21,\"p\":73}"},"fields":{"title":"The Nonzero Multiplicative Group Gives a p−1 Power Identity","stem":"Use the complete prime field Z_73, with prime p=73, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero element=21. Give (element,powers,last,is_one?,nonzero_multiplicative_group_cycle), listing exponents 1 through p−1.","answer":{"kind":"expression","value":"(21, [21, 3, 63, 9, 43, 27, 56, 8, 22, 24, 66, 72, 52, 70, 10, 64, 30, 46, 17, 65, 51, 49, 7, 1, 21, 3, 63, 9, 43, 27, 56, 8, 22, 24, 66, 72, 52, 70, 10, 64, 30, 46, 17, 65, 51, 49, 7, 1, 21, 3, 63, 9, 43, 27, 56, 8, 22, 24, 66, 72, 52, 70, 10, 64, 30, 46, 17, 65, 51, 49, 7, 1], 1, Yes, nonzero_multiplicative_group_cycle)","explanation":"Compute powers modulo p in exponent order. The final power equals one, but the element may have a smaller multiplicative order; p−1 is not asserted to be its exact order or a primitive-generator certificate. The exact result, retaining the requested order, is (21, [21, 3, 63, 9, 43, 27, 56, 8, 22, 24, 66, 72, 52, 70, 10, 64, 30, 46, 17, 65, 51, 49, 7, 1, 21, 3, 63, 9, 43, 27, 56, 8, 22, 24, 66, 72, 52, 70, 10, 64, 30, 46, 17, 65, 51, 49, 7, 1, 21, 3, 63, 9, 43, 27, 56, 8, 22, 24, 66, 72, 52, 70, 10, 64, 30, 46, 17, 65, 51, 49, 7, 1], 1, Yes, nonzero_multiplicative_group_cycle)."},"skills":["The Nonzero Multiplicative Group Gives a p−1 Power Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0047","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4801a02de5c0dedc354459a15f4c191f9d0ba599862401782e9c07182b288a3d","family_id":"qf-mod31-fielddeep-power-cycle","seed":202610316047,"semantic_template":"qf-mod31-fielddeep-power-cycle/{\"element\":22,\"p\":79}"},"fields":{"title":"The Nonzero Multiplicative Group Gives a p−1 Power Identity","stem":"Use the complete prime field Z_79, with prime p=79, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero element=22. Give (element,powers,last,is_one?,nonzero_multiplicative_group_cycle), listing exponents 1 through p−1.","answer":{"kind":"expression","value":"(22, [22, 10, 62, 21, 67, 52, 38, 46, 64, 65, 8, 18, 1, 22, 10, 62, 21, 67, 52, 38, 46, 64, 65, 8, 18, 1, 22, 10, 62, 21, 67, 52, 38, 46, 64, 65, 8, 18, 1, 22, 10, 62, 21, 67, 52, 38, 46, 64, 65, 8, 18, 1, 22, 10, 62, 21, 67, 52, 38, 46, 64, 65, 8, 18, 1, 22, 10, 62, 21, 67, 52, 38, 46, 64, 65, 8, 18, 1], 1, Yes, nonzero_multiplicative_group_cycle)","explanation":"Compute powers modulo p in exponent order. The final power equals one, but the element may have a smaller multiplicative order; p−1 is not asserted to be its exact order or a primitive-generator certificate. The exact result, retaining the requested order, is (22, [22, 10, 62, 21, 67, 52, 38, 46, 64, 65, 8, 18, 1, 22, 10, 62, 21, 67, 52, 38, 46, 64, 65, 8, 18, 1, 22, 10, 62, 21, 67, 52, 38, 46, 64, 65, 8, 18, 1, 22, 10, 62, 21, 67, 52, 38, 46, 64, 65, 8, 18, 1, 22, 10, 62, 21, 67, 52, 38, 46, 64, 65, 8, 18, 1, 22, 10, 62, 21, 67, 52, 38, 46, 64, 65, 8, 18, 1], 1, Yes, nonzero_multiplicative_group_cycle)."},"skills":["The Nonzero Multiplicative Group Gives a p−1 Power Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0048","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"67ffe0c1de12eea84d6db7b43bd1e46b255fbca1a493a30b28c075e1d3ad3495","family_id":"qf-mod31-fielddeep-power-cycle","seed":202610316048,"semantic_template":"qf-mod31-fielddeep-power-cycle/{\"element\":23,\"p\":83}"},"fields":{"title":"The Nonzero Multiplicative Group Gives a p−1 Power Identity","stem":"Use the complete prime field Z_83, with prime p=83, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero element=23. Give (element,powers,last,is_one?,nonzero_multiplicative_group_cycle), listing exponents 1 through p−1.","answer":{"kind":"expression","value":"(23, [23, 31, 49, 48, 25, 77, 28, 63, 38, 44, 16, 36, 81, 37, 21, 68, 70, 33, 12, 27, 40, 7, 78, 51, 11, 4, 9, 41, 30, 26, 17, 59, 29, 3, 69, 10, 64, 61, 75, 65, 1, 23, 31, 49, 48, 25, 77, 28, 63, 38, 44, 16, 36, 81, 37, 21, 68, 70, 33, 12, 27, 40, 7, 78, 51, 11, 4, 9, 41, 30, 26, 17, 59, 29, 3, 69, 10, 64, 61, 75, 65, 1], 1, Yes, nonzero_multiplicative_group_cycle)","explanation":"Compute powers modulo p in exponent order. The final power equals one, but the element may have a smaller multiplicative order; p−1 is not asserted to be its exact order or a primitive-generator certificate. The exact result, retaining the requested order, is (23, [23, 31, 49, 48, 25, 77, 28, 63, 38, 44, 16, 36, 81, 37, 21, 68, 70, 33, 12, 27, 40, 7, 78, 51, 11, 4, 9, 41, 30, 26, 17, 59, 29, 3, 69, 10, 64, 61, 75, 65, 1, 23, 31, 49, 48, 25, 77, 28, 63, 38, 44, 16, 36, 81, 37, 21, 68, 70, 33, 12, 27, 40, 7, 78, 51, 11, 4, 9, 41, 30, 26, 17, 59, 29, 3, 69, 10, 64, 61, 75, 65, 1], 1, Yes, nonzero_multiplicative_group_cycle)."},"skills":["The Nonzero Multiplicative Group Gives a p−1 Power Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0049","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ab463886b581c54e139bf7263c06f174d48b38eb8c86867e73a28666f7df9d7c","family_id":"qf-mod31-fielddeep-power-cycle","seed":202610316049,"semantic_template":"qf-mod31-fielddeep-power-cycle/{\"element\":24,\"p\":89}"},"fields":{"title":"The Nonzero Multiplicative Group Gives a p−1 Power Identity","stem":"Use the complete prime field Z_89, with prime p=89, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero element=24. Give (element,powers,last,is_one?,nonzero_multiplicative_group_cycle), listing exponents 1 through p−1.","answer":{"kind":"expression","value":"(24, [24, 42, 29, 73, 61, 40, 70, 78, 3, 72, 37, 87, 41, 5, 31, 32, 56, 9, 38, 22, 83, 34, 15, 4, 7, 79, 27, 25, 66, 71, 13, 45, 12, 21, 59, 81, 75, 20, 35, 39, 46, 36, 63, 88, 65, 47, 60, 16, 28, 49, 19, 11, 86, 17, 52, 2, 48, 84, 58, 57, 33, 80, 51, 67, 6, 55, 74, 85, 82, 10, 62, 64, 23, 18, 76, 44, 77, 68, 30, 8, 14, 69, 54, 50, 43, 53, 26, 1], 1, Yes, nonzero_multiplicative_group_cycle)","explanation":"Compute powers modulo p in exponent order. The final power equals one, but the element may have a smaller multiplicative order; p−1 is not asserted to be its exact order or a primitive-generator certificate. The exact result, retaining the requested order, is (24, [24, 42, 29, 73, 61, 40, 70, 78, 3, 72, 37, 87, 41, 5, 31, 32, 56, 9, 38, 22, 83, 34, 15, 4, 7, 79, 27, 25, 66, 71, 13, 45, 12, 21, 59, 81, 75, 20, 35, 39, 46, 36, 63, 88, 65, 47, 60, 16, 28, 49, 19, 11, 86, 17, 52, 2, 48, 84, 58, 57, 33, 80, 51, 67, 6, 55, 74, 85, 82, 10, 62, 64, 23, 18, 76, 44, 77, 68, 30, 8, 14, 69, 54, 50, 43, 53, 26, 1], 1, Yes, nonzero_multiplicative_group_cycle)."},"skills":["The Nonzero Multiplicative Group Gives a p−1 Power Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0050","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f2393607407dd9888ac6c912c6bf075bdd1396a5c3ba307915f2cdad3758a7dc","family_id":"qf-mod31-fielddeep-power-cycle","seed":202610316050,"semantic_template":"qf-mod31-fielddeep-power-cycle/{\"element\":25,\"p\":97}"},"fields":{"title":"The Nonzero Multiplicative Group Gives a p−1 Power Identity","stem":"Use the complete prime field Z_97, with prime p=97, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero element=25. Give (element,powers,last,is_one?,nonzero_multiplicative_group_cycle), listing exponents 1 through p−1.","answer":{"kind":"expression","value":"(25, [25, 43, 8, 6, 53, 64, 48, 36, 27, 93, 94, 22, 65, 73, 79, 35, 2, 50, 86, 16, 12, 9, 31, 96, 72, 54, 89, 91, 44, 33, 49, 61, 70, 4, 3, 75, 32, 24, 18, 62, 95, 47, 11, 81, 85, 88, 66, 1, 25, 43, 8, 6, 53, 64, 48, 36, 27, 93, 94, 22, 65, 73, 79, 35, 2, 50, 86, 16, 12, 9, 31, 96, 72, 54, 89, 91, 44, 33, 49, 61, 70, 4, 3, 75, 32, 24, 18, 62, 95, 47, 11, 81, 85, 88, 66, 1], 1, Yes, nonzero_multiplicative_group_cycle)","explanation":"Compute powers modulo p in exponent order. The final power equals one, but the element may have a smaller multiplicative order; p−1 is not asserted to be its exact order or a primitive-generator certificate. The exact result, retaining the requested order, is (25, [25, 43, 8, 6, 53, 64, 48, 36, 27, 93, 94, 22, 65, 73, 79, 35, 2, 50, 86, 16, 12, 9, 31, 96, 72, 54, 89, 91, 44, 33, 49, 61, 70, 4, 3, 75, 32, 24, 18, 62, 95, 47, 11, 81, 85, 88, 66, 1, 25, 43, 8, 6, 53, 64, 48, 36, 27, 93, 94, 22, 65, 73, 79, 35, 2, 50, 86, 16, 12, 9, 31, 96, 72, 54, 89, 91, 44, 33, 49, 61, 70, 4, 3, 75, 32, 24, 18, 62, 95, 47, 11, 81, 85, 88, 66, 1], 1, Yes, nonzero_multiplicative_group_cycle)."},"skills":["The Nonzero Multiplicative Group Gives a p−1 Power Identity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0051","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b99e31138d4eab9718d8d206415403797f61d98ebf9262f3c267098deca74601","family_id":"qf-mod31-fielddeep-affine-solve","seed":202610316051,"semantic_template":"qf-mod31-fielddeep-affine-solve/{\"a\":1,\"b\":1,\"c\":1,\"p\":2}"},"fields":{"title":"Solve a Nonzero-Coefficient Affine Equation in the Prime Field","stem":"Use the complete prime field Z_2, with prime p=2, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero a=1, b=1, c=1; solve a*x+b=c. Give (a,b,c,inverse,solution,substitution,unique?).","answer":{"kind":"expression","value":"(1, 1, 1, 1, 0, 1, Yes)","explanation":"Invert a modulo p, multiply c−b by that inverse, and substitute back under the same field operations. Nonzero multiplication is bijective, so the solution is unique. The exact result, retaining the requested order, is (1, 1, 1, 1, 0, 1, Yes)."},"skills":["Solve a Nonzero-Coefficient Affine Equation in the Prime Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0052","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0d45e3a3a976f099e1aceb87bf85e99509baa7c6baee200e155b946d74e29ecd","family_id":"qf-mod31-fielddeep-affine-solve","seed":202610316052,"semantic_template":"qf-mod31-fielddeep-affine-solve/{\"a\":2,\"b\":2,\"c\":0,\"p\":3}"},"fields":{"title":"Solve a Nonzero-Coefficient Affine Equation in the Prime Field","stem":"Use the complete prime field Z_3, with prime p=3, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero a=2, b=2, c=0; solve a*x+b=c. Give (a,b,c,inverse,solution,substitution,unique?).","answer":{"kind":"expression","value":"(2, 2, 0, 2, 2, 0, Yes)","explanation":"Invert a modulo p, multiply c−b by that inverse, and substitute back under the same field operations. Nonzero multiplication is bijective, so the solution is unique. The exact result, retaining the requested order, is (2, 2, 0, 2, 2, 0, Yes)."},"skills":["Solve a Nonzero-Coefficient Affine Equation in the Prime Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0053","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a813ab5753fb622dd6399d17900362720eb1fe6b15a193c8a34d38c18cbfba6b","family_id":"qf-mod31-fielddeep-affine-solve","seed":202610316053,"semantic_template":"qf-mod31-fielddeep-affine-solve/{\"a\":3,\"b\":3,\"c\":0,\"p\":5}"},"fields":{"title":"Solve a Nonzero-Coefficient Affine Equation in the Prime Field","stem":"Use the complete prime field Z_5, with prime p=5, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero a=3, b=3, c=0; solve a*x+b=c. Give (a,b,c,inverse,solution,substitution,unique?).","answer":{"kind":"expression","value":"(3, 3, 0, 2, 4, 0, Yes)","explanation":"Invert a modulo p, multiply c−b by that inverse, and substitute back under the same field operations. Nonzero multiplication is bijective, so the solution is unique. The exact result, retaining the requested order, is (3, 3, 0, 2, 4, 0, Yes)."},"skills":["Solve a Nonzero-Coefficient Affine Equation in the Prime Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0054","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0a625cc773ad810d1a45ea9b005257a53c201102de61c056ab329d6b6afbac70","family_id":"qf-mod31-fielddeep-affine-solve","seed":202610316054,"semantic_template":"qf-mod31-fielddeep-affine-solve/{\"a\":4,\"b\":4,\"c\":0,\"p\":7}"},"fields":{"title":"Solve a Nonzero-Coefficient Affine Equation in the Prime Field","stem":"Use the complete prime field Z_7, with prime p=7, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero a=4, b=4, c=0; solve a*x+b=c. Give (a,b,c,inverse,solution,substitution,unique?).","answer":{"kind":"expression","value":"(4, 4, 0, 2, 6, 0, Yes)","explanation":"Invert a modulo p, multiply c−b by that inverse, and substitute back under the same field operations. Nonzero multiplication is bijective, so the solution is unique. The exact result, retaining the requested order, is (4, 4, 0, 2, 6, 0, Yes)."},"skills":["Solve a Nonzero-Coefficient Affine Equation in the Prime Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0055","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7c68ea51618d662fb0f265beaa699c53578ed8a04a47913e795a323cb6848766","family_id":"qf-mod31-fielddeep-affine-solve","seed":202610316055,"semantic_template":"qf-mod31-fielddeep-affine-solve/{\"a\":5,\"b\":5,\"c\":9,\"p\":11}"},"fields":{"title":"Solve a Nonzero-Coefficient Affine Equation in the Prime Field","stem":"Use the complete prime field Z_11, with prime p=11, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero a=5, b=5, c=9; solve a*x+b=c. Give (a,b,c,inverse,solution,substitution,unique?).","answer":{"kind":"expression","value":"(5, 5, 9, 9, 3, 9, Yes)","explanation":"Invert a modulo p, multiply c−b by that inverse, and substitute back under the same field operations. Nonzero multiplication is bijective, so the solution is unique. The exact result, retaining the requested order, is (5, 5, 9, 9, 3, 9, Yes)."},"skills":["Solve a Nonzero-Coefficient Affine Equation in the Prime Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0056","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b32d0938de030cb7076c7eee0c8a2a7c4d97c831cfd2b2f4b38ae854d7d45b4","family_id":"qf-mod31-fielddeep-affine-solve","seed":202610316056,"semantic_template":"qf-mod31-fielddeep-affine-solve/{\"a\":6,\"b\":6,\"c\":11,\"p\":13}"},"fields":{"title":"Solve a Nonzero-Coefficient Affine Equation in the Prime Field","stem":"Use the complete prime field Z_13, with prime p=13, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero a=6, b=6, c=11; solve a*x+b=c. Give (a,b,c,inverse,solution,substitution,unique?).","answer":{"kind":"expression","value":"(6, 6, 11, 11, 3, 11, Yes)","explanation":"Invert a modulo p, multiply c−b by that inverse, and substitute back under the same field operations. Nonzero multiplication is bijective, so the solution is unique. The exact result, retaining the requested order, is (6, 6, 11, 11, 3, 11, Yes)."},"skills":["Solve a Nonzero-Coefficient Affine Equation in the Prime Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0057","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c681abf43dcfa625218d71e1ffd7a45e9fc40435b1bd7d23ca9230b161d4825b","family_id":"qf-mod31-fielddeep-affine-solve","seed":202610316057,"semantic_template":"qf-mod31-fielddeep-affine-solve/{\"a\":7,\"b\":7,\"c\":13,\"p\":17}"},"fields":{"title":"Solve a Nonzero-Coefficient Affine Equation in the Prime Field","stem":"Use the complete prime field Z_17, with prime p=17, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero a=7, b=7, c=13; solve a*x+b=c. Give (a,b,c,inverse,solution,substitution,unique?).","answer":{"kind":"expression","value":"(7, 7, 13, 5, 13, 13, Yes)","explanation":"Invert a modulo p, multiply c−b by that inverse, and substitute back under the same field operations. Nonzero multiplication is bijective, so the solution is unique. The exact result, retaining the requested order, is (7, 7, 13, 5, 13, 13, Yes)."},"skills":["Solve a Nonzero-Coefficient Affine Equation in the Prime Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0058","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ad9b7b05c03e78bab73d627a7a9967c52ed1ac6aa9297538889da215207c805","family_id":"qf-mod31-fielddeep-affine-solve","seed":202610316058,"semantic_template":"qf-mod31-fielddeep-affine-solve/{\"a\":8,\"b\":8,\"c\":15,\"p\":19}"},"fields":{"title":"Solve a Nonzero-Coefficient Affine Equation in the Prime Field","stem":"Use the complete prime field Z_19, with prime p=19, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero a=8, b=8, c=15; solve a*x+b=c. Give (a,b,c,inverse,solution,substitution,unique?).","answer":{"kind":"expression","value":"(8, 8, 15, 12, 8, 15, Yes)","explanation":"Invert a modulo p, multiply c−b by that inverse, and substitute back under the same field operations. Nonzero multiplication is bijective, so the solution is unique. The exact result, retaining the requested order, is (8, 8, 15, 12, 8, 15, Yes)."},"skills":["Solve a Nonzero-Coefficient Affine Equation in the Prime Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0059","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"39327bf4b4ff1964b02d3d028a189cbcc9f660dde82d722dc0a6ba6919d455d2","family_id":"qf-mod31-fielddeep-affine-solve","seed":202610316059,"semantic_template":"qf-mod31-fielddeep-affine-solve/{\"a\":9,\"b\":9,\"c\":17,\"p\":23}"},"fields":{"title":"Solve a Nonzero-Coefficient Affine Equation in the Prime Field","stem":"Use the complete prime field Z_23, with prime p=23, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero a=9, b=9, c=17; solve a*x+b=c. Give (a,b,c,inverse,solution,substitution,unique?).","answer":{"kind":"expression","value":"(9, 9, 17, 18, 6, 17, Yes)","explanation":"Invert a modulo p, multiply c−b by that inverse, and substitute back under the same field operations. Nonzero multiplication is bijective, so the solution is unique. The exact result, retaining the requested order, is (9, 9, 17, 18, 6, 17, Yes)."},"skills":["Solve a Nonzero-Coefficient Affine Equation in the Prime Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0060","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7c54dbaffd59ff696da4090e6c7be71c5f49f222fc3ba8264f3aea118c6960b9","family_id":"qf-mod31-fielddeep-affine-solve","seed":202610316060,"semantic_template":"qf-mod31-fielddeep-affine-solve/{\"a\":10,\"b\":10,\"c\":19,\"p\":29}"},"fields":{"title":"Solve a Nonzero-Coefficient Affine Equation in the Prime Field","stem":"Use the complete prime field Z_29, with prime p=29, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero a=10, b=10, c=19; solve a*x+b=c. Give (a,b,c,inverse,solution,substitution,unique?).","answer":{"kind":"expression","value":"(10, 10, 19, 3, 27, 19, Yes)","explanation":"Invert a modulo p, multiply c−b by that inverse, and substitute back under the same field operations. Nonzero multiplication is bijective, so the solution is unique. The exact result, retaining the requested order, is (10, 10, 19, 3, 27, 19, Yes)."},"skills":["Solve a Nonzero-Coefficient Affine Equation in the Prime Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0061","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5bf1311a6296d0b0a4f5422a638062dba09ddb4a066b1887469645801d6bd927","family_id":"qf-mod31-fielddeep-affine-solve","seed":202610316061,"semantic_template":"qf-mod31-fielddeep-affine-solve/{\"a\":11,\"b\":11,\"c\":21,\"p\":31}"},"fields":{"title":"Solve a Nonzero-Coefficient Affine Equation in the Prime Field","stem":"Use the complete prime field Z_31, with prime p=31, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero a=11, b=11, c=21; solve a*x+b=c. Give (a,b,c,inverse,solution,substitution,unique?).","answer":{"kind":"expression","value":"(11, 11, 21, 17, 15, 21, Yes)","explanation":"Invert a modulo p, multiply c−b by that inverse, and substitute back under the same field operations. Nonzero multiplication is bijective, so the solution is unique. The exact result, retaining the requested order, is (11, 11, 21, 17, 15, 21, Yes)."},"skills":["Solve a Nonzero-Coefficient Affine Equation in the Prime Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0062","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"067f4a7ae39fa581e262ae63ac9867def2fd7b33b79a352ef25063cf9c9507ce","family_id":"qf-mod31-fielddeep-affine-solve","seed":202610316062,"semantic_template":"qf-mod31-fielddeep-affine-solve/{\"a\":12,\"b\":12,\"c\":23,\"p\":37}"},"fields":{"title":"Solve a Nonzero-Coefficient Affine Equation in the Prime Field","stem":"Use the complete prime field Z_37, with prime p=37, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero a=12, b=12, c=23; solve a*x+b=c. Give (a,b,c,inverse,solution,substitution,unique?).","answer":{"kind":"expression","value":"(12, 12, 23, 34, 4, 23, Yes)","explanation":"Invert a modulo p, multiply c−b by that inverse, and substitute back under the same field operations. Nonzero multiplication is bijective, so the solution is unique. The exact result, retaining the requested order, is (12, 12, 23, 34, 4, 23, Yes)."},"skills":["Solve a Nonzero-Coefficient Affine Equation in the Prime Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0063","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5eb48ee70f86710544a02f162f9a4a93c12e6c1753d8b1625898a217f9b3fccc","family_id":"qf-mod31-fielddeep-affine-solve","seed":202610316063,"semantic_template":"qf-mod31-fielddeep-affine-solve/{\"a\":13,\"b\":13,\"c\":25,\"p\":41}"},"fields":{"title":"Solve a Nonzero-Coefficient Affine Equation in the Prime Field","stem":"Use the complete prime field Z_41, with prime p=41, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero a=13, b=13, c=25; solve a*x+b=c. Give (a,b,c,inverse,solution,substitution,unique?).","answer":{"kind":"expression","value":"(13, 13, 25, 19, 23, 25, Yes)","explanation":"Invert a modulo p, multiply c−b by that inverse, and substitute back under the same field operations. Nonzero multiplication is bijective, so the solution is unique. The exact result, retaining the requested order, is (13, 13, 25, 19, 23, 25, Yes)."},"skills":["Solve a Nonzero-Coefficient Affine Equation in the Prime Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0064","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"50cf95b87beb4a21fd639aefdcfbd04b9ac8e84e8f4c79537e92383795cb2181","family_id":"qf-mod31-fielddeep-affine-solve","seed":202610316064,"semantic_template":"qf-mod31-fielddeep-affine-solve/{\"a\":14,\"b\":14,\"c\":27,\"p\":43}"},"fields":{"title":"Solve a Nonzero-Coefficient Affine Equation in the Prime Field","stem":"Use the complete prime field Z_43, with prime p=43, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero a=14, b=14, c=27; solve a*x+b=c. Give (a,b,c,inverse,solution,substitution,unique?).","answer":{"kind":"expression","value":"(14, 14, 27, 40, 4, 27, Yes)","explanation":"Invert a modulo p, multiply c−b by that inverse, and substitute back under the same field operations. Nonzero multiplication is bijective, so the solution is unique. The exact result, retaining the requested order, is (14, 14, 27, 40, 4, 27, Yes)."},"skills":["Solve a Nonzero-Coefficient Affine Equation in the Prime Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0065","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"682e01a5203aa94d5f0197eaa7117b83736fb0ee114d4d81d123436fea8150de","family_id":"qf-mod31-fielddeep-affine-solve","seed":202610316065,"semantic_template":"qf-mod31-fielddeep-affine-solve/{\"a\":15,\"b\":15,\"c\":29,\"p\":47}"},"fields":{"title":"Solve a Nonzero-Coefficient Affine Equation in the Prime Field","stem":"Use the complete prime field Z_47, with prime p=47, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero a=15, b=15, c=29; solve a*x+b=c. Give (a,b,c,inverse,solution,substitution,unique?).","answer":{"kind":"expression","value":"(15, 15, 29, 22, 26, 29, Yes)","explanation":"Invert a modulo p, multiply c−b by that inverse, and substitute back under the same field operations. Nonzero multiplication is bijective, so the solution is unique. The exact result, retaining the requested order, is (15, 15, 29, 22, 26, 29, Yes)."},"skills":["Solve a Nonzero-Coefficient Affine Equation in the Prime Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0066","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bef668dafc31c9a648c4f817a2e5b56c7fbb8ac173c5592785d7378c320e77cd","family_id":"qf-mod31-fielddeep-affine-solve","seed":202610316066,"semantic_template":"qf-mod31-fielddeep-affine-solve/{\"a\":16,\"b\":16,\"c\":31,\"p\":53}"},"fields":{"title":"Solve a Nonzero-Coefficient Affine Equation in the Prime Field","stem":"Use the complete prime field Z_53, with prime p=53, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero a=16, b=16, c=31; solve a*x+b=c. Give (a,b,c,inverse,solution,substitution,unique?).","answer":{"kind":"expression","value":"(16, 16, 31, 10, 44, 31, Yes)","explanation":"Invert a modulo p, multiply c−b by that inverse, and substitute back under the same field operations. Nonzero multiplication is bijective, so the solution is unique. The exact result, retaining the requested order, is (16, 16, 31, 10, 44, 31, Yes)."},"skills":["Solve a Nonzero-Coefficient Affine Equation in the Prime Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0067","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ce18d8a95ac54436923dc70b275b4d5c270859daa41c4d21527d2bec0e588a21","family_id":"qf-mod31-fielddeep-affine-solve","seed":202610316067,"semantic_template":"qf-mod31-fielddeep-affine-solve/{\"a\":17,\"b\":17,\"c\":33,\"p\":59}"},"fields":{"title":"Solve a Nonzero-Coefficient Affine Equation in the Prime Field","stem":"Use the complete prime field Z_59, with prime p=59, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero a=17, b=17, c=33; solve a*x+b=c. Give (a,b,c,inverse,solution,substitution,unique?).","answer":{"kind":"expression","value":"(17, 17, 33, 7, 53, 33, Yes)","explanation":"Invert a modulo p, multiply c−b by that inverse, and substitute back under the same field operations. Nonzero multiplication is bijective, so the solution is unique. The exact result, retaining the requested order, is (17, 17, 33, 7, 53, 33, Yes)."},"skills":["Solve a Nonzero-Coefficient Affine Equation in the Prime Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0068","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f8c60bbfc75566a849db007cf8feaff9ec6aa87a205da36c859ae8bfe1764f9c","family_id":"qf-mod31-fielddeep-affine-solve","seed":202610316068,"semantic_template":"qf-mod31-fielddeep-affine-solve/{\"a\":18,\"b\":18,\"c\":35,\"p\":61}"},"fields":{"title":"Solve a Nonzero-Coefficient Affine Equation in the Prime Field","stem":"Use the complete prime field Z_61, with prime p=61, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero a=18, b=18, c=35; solve a*x+b=c. Give (a,b,c,inverse,solution,substitution,unique?).","answer":{"kind":"expression","value":"(18, 18, 35, 17, 45, 35, Yes)","explanation":"Invert a modulo p, multiply c−b by that inverse, and substitute back under the same field operations. Nonzero multiplication is bijective, so the solution is unique. The exact result, retaining the requested order, is (18, 18, 35, 17, 45, 35, Yes)."},"skills":["Solve a Nonzero-Coefficient Affine Equation in the Prime Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0069","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"111fdf29ea3e51e27c121bbfa8d44b8f46b69dfc04b6ed7478a18ed241417467","family_id":"qf-mod31-fielddeep-affine-solve","seed":202610316069,"semantic_template":"qf-mod31-fielddeep-affine-solve/{\"a\":19,\"b\":19,\"c\":37,\"p\":67}"},"fields":{"title":"Solve a Nonzero-Coefficient Affine Equation in the Prime Field","stem":"Use the complete prime field Z_67, with prime p=67, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero a=19, b=19, c=37; solve a*x+b=c. Give (a,b,c,inverse,solution,substitution,unique?).","answer":{"kind":"expression","value":"(19, 19, 37, 60, 8, 37, Yes)","explanation":"Invert a modulo p, multiply c−b by that inverse, and substitute back under the same field operations. Nonzero multiplication is bijective, so the solution is unique. The exact result, retaining the requested order, is (19, 19, 37, 60, 8, 37, Yes)."},"skills":["Solve a Nonzero-Coefficient Affine Equation in the Prime Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0070","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0ce6690ebb8e3842a85d439f60d42e1704f2be6fafe5a9fdaf8a89831873e784","family_id":"qf-mod31-fielddeep-affine-solve","seed":202610316070,"semantic_template":"qf-mod31-fielddeep-affine-solve/{\"a\":20,\"b\":20,\"c\":39,\"p\":71}"},"fields":{"title":"Solve a Nonzero-Coefficient Affine Equation in the Prime Field","stem":"Use the complete prime field Z_71, with prime p=71, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero a=20, b=20, c=39; solve a*x+b=c. Give (a,b,c,inverse,solution,substitution,unique?).","answer":{"kind":"expression","value":"(20, 20, 39, 32, 40, 39, Yes)","explanation":"Invert a modulo p, multiply c−b by that inverse, and substitute back under the same field operations. Nonzero multiplication is bijective, so the solution is unique. The exact result, retaining the requested order, is (20, 20, 39, 32, 40, 39, Yes)."},"skills":["Solve a Nonzero-Coefficient Affine Equation in the Prime Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0071","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b28ad72aeb32fe9781942548f75c59063de343b4811df05d40ed53928499676d","family_id":"qf-mod31-fielddeep-affine-solve","seed":202610316071,"semantic_template":"qf-mod31-fielddeep-affine-solve/{\"a\":21,\"b\":21,\"c\":41,\"p\":73}"},"fields":{"title":"Solve a Nonzero-Coefficient Affine Equation in the Prime Field","stem":"Use the complete prime field Z_73, with prime p=73, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero a=21, b=21, c=41; solve a*x+b=c. Give (a,b,c,inverse,solution,substitution,unique?).","answer":{"kind":"expression","value":"(21, 21, 41, 7, 67, 41, Yes)","explanation":"Invert a modulo p, multiply c−b by that inverse, and substitute back under the same field operations. Nonzero multiplication is bijective, so the solution is unique. The exact result, retaining the requested order, is (21, 21, 41, 7, 67, 41, Yes)."},"skills":["Solve a Nonzero-Coefficient Affine Equation in the Prime Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0072","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"176f1c0aa954ec1f0de42f17fd04c1f7c84f88f873f188a728618459c3ab8d9f","family_id":"qf-mod31-fielddeep-affine-solve","seed":202610316072,"semantic_template":"qf-mod31-fielddeep-affine-solve/{\"a\":22,\"b\":22,\"c\":43,\"p\":79}"},"fields":{"title":"Solve a Nonzero-Coefficient Affine Equation in the Prime Field","stem":"Use the complete prime field Z_79, with prime p=79, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero a=22, b=22, c=43; solve a*x+b=c. Give (a,b,c,inverse,solution,substitution,unique?).","answer":{"kind":"expression","value":"(22, 22, 43, 18, 62, 43, Yes)","explanation":"Invert a modulo p, multiply c−b by that inverse, and substitute back under the same field operations. Nonzero multiplication is bijective, so the solution is unique. The exact result, retaining the requested order, is (22, 22, 43, 18, 62, 43, Yes)."},"skills":["Solve a Nonzero-Coefficient Affine Equation in the Prime Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0073","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"187267aece4206288ce45dee5b6348905e797d9806e5fc2651f723335076682a","family_id":"qf-mod31-fielddeep-affine-solve","seed":202610316073,"semantic_template":"qf-mod31-fielddeep-affine-solve/{\"a\":23,\"b\":23,\"c\":45,\"p\":83}"},"fields":{"title":"Solve a Nonzero-Coefficient Affine Equation in the Prime Field","stem":"Use the complete prime field Z_83, with prime p=83, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero a=23, b=23, c=45; solve a*x+b=c. Give (a,b,c,inverse,solution,substitution,unique?).","answer":{"kind":"expression","value":"(23, 23, 45, 65, 19, 45, Yes)","explanation":"Invert a modulo p, multiply c−b by that inverse, and substitute back under the same field operations. Nonzero multiplication is bijective, so the solution is unique. The exact result, retaining the requested order, is (23, 23, 45, 65, 19, 45, Yes)."},"skills":["Solve a Nonzero-Coefficient Affine Equation in the Prime Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0074","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"79a02bf91611084e796f2b56e4fac0b0512bb1cb862a5969c5395bccffda656e","family_id":"qf-mod31-fielddeep-affine-solve","seed":202610316074,"semantic_template":"qf-mod31-fielddeep-affine-solve/{\"a\":24,\"b\":24,\"c\":47,\"p\":89}"},"fields":{"title":"Solve a Nonzero-Coefficient Affine Equation in the Prime Field","stem":"Use the complete prime field Z_89, with prime p=89, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero a=24, b=24, c=47; solve a*x+b=c. Give (a,b,c,inverse,solution,substitution,unique?).","answer":{"kind":"expression","value":"(24, 24, 47, 26, 64, 47, Yes)","explanation":"Invert a modulo p, multiply c−b by that inverse, and substitute back under the same field operations. Nonzero multiplication is bijective, so the solution is unique. The exact result, retaining the requested order, is (24, 24, 47, 26, 64, 47, Yes)."},"skills":["Solve a Nonzero-Coefficient Affine Equation in the Prime Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0075","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5e45f27557aed6010aa232c50fe4c9503e8619963aadeeb7c0ca1129b98ee2e2","family_id":"qf-mod31-fielddeep-affine-solve","seed":202610316075,"semantic_template":"qf-mod31-fielddeep-affine-solve/{\"a\":25,\"b\":25,\"c\":49,\"p\":97}"},"fields":{"title":"Solve a Nonzero-Coefficient Affine Equation in the Prime Field","stem":"Use the complete prime field Z_97, with prime p=97, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. Nonzero a=25, b=25, c=49; solve a*x+b=c. Give (a,b,c,inverse,solution,substitution,unique?).","answer":{"kind":"expression","value":"(25, 25, 49, 66, 32, 49, Yes)","explanation":"Invert a modulo p, multiply c−b by that inverse, and substitute back under the same field operations. Nonzero multiplication is bijective, so the solution is unique. The exact result, retaining the requested order, is (25, 25, 49, 66, 32, 49, Yes)."},"skills":["Solve a Nonzero-Coefficient Affine Equation in the Prime Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0076","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b3a482f5046e1c7f477daf538e4cddc0418b136ad6f536698e2bb06c756ae341","family_id":"qf-mod31-fielddeep-frobenius","seed":202610316076,"semantic_template":"qf-mod31-fielddeep-frobenius/{\"element\":0,\"p\":2}"},"fields":{"title":"Frobenius Fixes Every Element of This Prime Field","stem":"Use the complete prime field Z_2, with prime p=2, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. element=0, possibly zero. Give (element,p,power,equal?,Frobenius_on_prime_field), with power=element^p modulo p.","answer":{"kind":"expression","value":"(0, 2, 0, Yes, Frobenius_on_prime_field)","explanation":"Compute the p-th power in the specified prime field. This includes zero and does not assert that x^p=x holds in every finite extension field. The exact result, retaining the requested order, is (0, 2, 0, Yes, Frobenius_on_prime_field)."},"skills":["Frobenius Fixes Every Element of This Prime Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0077","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"01864d438b3cec66ef0d199739bc94ba66c21c1dd18c89ce2280aac4c7d100eb","family_id":"qf-mod31-fielddeep-frobenius","seed":202610316077,"semantic_template":"qf-mod31-fielddeep-frobenius/{\"element\":1,\"p\":3}"},"fields":{"title":"Frobenius Fixes Every Element of This Prime Field","stem":"Use the complete prime field Z_3, with prime p=3, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. element=1, possibly zero. Give (element,p,power,equal?,Frobenius_on_prime_field), with power=element^p modulo p.","answer":{"kind":"expression","value":"(1, 3, 1, Yes, Frobenius_on_prime_field)","explanation":"Compute the p-th power in the specified prime field. This includes zero and does not assert that x^p=x holds in every finite extension field. The exact result, retaining the requested order, is (1, 3, 1, Yes, Frobenius_on_prime_field)."},"skills":["Frobenius Fixes Every Element of This Prime Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0078","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1c6e6ffaa488205dab0825128acf2ff3a312204e3bb441d9cfa49b625cba46cc","family_id":"qf-mod31-fielddeep-frobenius","seed":202610316078,"semantic_template":"qf-mod31-fielddeep-frobenius/{\"element\":2,\"p\":5}"},"fields":{"title":"Frobenius Fixes Every Element of This Prime Field","stem":"Use the complete prime field Z_5, with prime p=5, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. element=2, possibly zero. Give (element,p,power,equal?,Frobenius_on_prime_field), with power=element^p modulo p.","answer":{"kind":"expression","value":"(2, 5, 2, Yes, Frobenius_on_prime_field)","explanation":"Compute the p-th power in the specified prime field. This includes zero and does not assert that x^p=x holds in every finite extension field. The exact result, retaining the requested order, is (2, 5, 2, Yes, Frobenius_on_prime_field)."},"skills":["Frobenius Fixes Every Element of This Prime Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0079","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"897c4f611b10818b7c316d5f05138c14573ce0b5d7d6dda98829c692c263c1b1","family_id":"qf-mod31-fielddeep-frobenius","seed":202610316079,"semantic_template":"qf-mod31-fielddeep-frobenius/{\"element\":3,\"p\":7}"},"fields":{"title":"Frobenius Fixes Every Element of This Prime Field","stem":"Use the complete prime field Z_7, with prime p=7, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. element=3, possibly zero. Give (element,p,power,equal?,Frobenius_on_prime_field), with power=element^p modulo p.","answer":{"kind":"expression","value":"(3, 7, 3, Yes, Frobenius_on_prime_field)","explanation":"Compute the p-th power in the specified prime field. This includes zero and does not assert that x^p=x holds in every finite extension field. The exact result, retaining the requested order, is (3, 7, 3, Yes, Frobenius_on_prime_field)."},"skills":["Frobenius Fixes Every Element of This Prime Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0080","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"45754889146565fb85e5643f3c18cdaae6b5b9e0952c8455145744234be3b5b0","family_id":"qf-mod31-fielddeep-frobenius","seed":202610316080,"semantic_template":"qf-mod31-fielddeep-frobenius/{\"element\":4,\"p\":11}"},"fields":{"title":"Frobenius Fixes Every Element of This Prime Field","stem":"Use the complete prime field Z_11, with prime p=11, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. element=4, possibly zero. Give (element,p,power,equal?,Frobenius_on_prime_field), with power=element^p modulo p.","answer":{"kind":"expression","value":"(4, 11, 4, Yes, Frobenius_on_prime_field)","explanation":"Compute the p-th power in the specified prime field. This includes zero and does not assert that x^p=x holds in every finite extension field. The exact result, retaining the requested order, is (4, 11, 4, Yes, Frobenius_on_prime_field)."},"skills":["Frobenius Fixes Every Element of This Prime Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0081","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"512cba0acf38c1511e2af6c26beac012c665a6d11f4e6edcfe0f4df4e5303af7","family_id":"qf-mod31-fielddeep-frobenius","seed":202610316081,"semantic_template":"qf-mod31-fielddeep-frobenius/{\"element\":5,\"p\":13}"},"fields":{"title":"Frobenius Fixes Every Element of This Prime Field","stem":"Use the complete prime field Z_13, with prime p=13, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. element=5, possibly zero. Give (element,p,power,equal?,Frobenius_on_prime_field), with power=element^p modulo p.","answer":{"kind":"expression","value":"(5, 13, 5, Yes, Frobenius_on_prime_field)","explanation":"Compute the p-th power in the specified prime field. This includes zero and does not assert that x^p=x holds in every finite extension field. The exact result, retaining the requested order, is (5, 13, 5, Yes, Frobenius_on_prime_field)."},"skills":["Frobenius Fixes Every Element of This Prime Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0082","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a96eb60b5b00b84a6a37b257bf75628d71d47d7570ba9bcc836a6b60f7f2b93e","family_id":"qf-mod31-fielddeep-frobenius","seed":202610316082,"semantic_template":"qf-mod31-fielddeep-frobenius/{\"element\":6,\"p\":17}"},"fields":{"title":"Frobenius Fixes Every Element of This Prime Field","stem":"Use the complete prime field Z_17, with prime p=17, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. element=6, possibly zero. Give (element,p,power,equal?,Frobenius_on_prime_field), with power=element^p modulo p.","answer":{"kind":"expression","value":"(6, 17, 6, Yes, Frobenius_on_prime_field)","explanation":"Compute the p-th power in the specified prime field. This includes zero and does not assert that x^p=x holds in every finite extension field. The exact result, retaining the requested order, is (6, 17, 6, Yes, Frobenius_on_prime_field)."},"skills":["Frobenius Fixes Every Element of This Prime Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0083","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"68918a83e63927bb24c62efb495692ce126f0e0bf7399cb8ef81ef903542643b","family_id":"qf-mod31-fielddeep-frobenius","seed":202610316083,"semantic_template":"qf-mod31-fielddeep-frobenius/{\"element\":7,\"p\":19}"},"fields":{"title":"Frobenius Fixes Every Element of This Prime Field","stem":"Use the complete prime field Z_19, with prime p=19, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. element=7, possibly zero. Give (element,p,power,equal?,Frobenius_on_prime_field), with power=element^p modulo p.","answer":{"kind":"expression","value":"(7, 19, 7, Yes, Frobenius_on_prime_field)","explanation":"Compute the p-th power in the specified prime field. This includes zero and does not assert that x^p=x holds in every finite extension field. The exact result, retaining the requested order, is (7, 19, 7, Yes, Frobenius_on_prime_field)."},"skills":["Frobenius Fixes Every Element of This Prime Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0084","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cc5f4d5eb39f86bdc5e8314f0d59e6dd234947eaa936422ce722296302419bc5","family_id":"qf-mod31-fielddeep-frobenius","seed":202610316084,"semantic_template":"qf-mod31-fielddeep-frobenius/{\"element\":8,\"p\":23}"},"fields":{"title":"Frobenius Fixes Every Element of This Prime Field","stem":"Use the complete prime field Z_23, with prime p=23, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. element=8, possibly zero. Give (element,p,power,equal?,Frobenius_on_prime_field), with power=element^p modulo p.","answer":{"kind":"expression","value":"(8, 23, 8, Yes, Frobenius_on_prime_field)","explanation":"Compute the p-th power in the specified prime field. This includes zero and does not assert that x^p=x holds in every finite extension field. The exact result, retaining the requested order, is (8, 23, 8, Yes, Frobenius_on_prime_field)."},"skills":["Frobenius Fixes Every Element of This Prime Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0085","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"44ab2264ec797508fd310d5c493323b23504ebded5491964b21cb8ccd9ecd02b","family_id":"qf-mod31-fielddeep-frobenius","seed":202610316085,"semantic_template":"qf-mod31-fielddeep-frobenius/{\"element\":9,\"p\":29}"},"fields":{"title":"Frobenius Fixes Every Element of This Prime Field","stem":"Use the complete prime field Z_29, with prime p=29, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. element=9, possibly zero. Give (element,p,power,equal?,Frobenius_on_prime_field), with power=element^p modulo p.","answer":{"kind":"expression","value":"(9, 29, 9, Yes, Frobenius_on_prime_field)","explanation":"Compute the p-th power in the specified prime field. This includes zero and does not assert that x^p=x holds in every finite extension field. The exact result, retaining the requested order, is (9, 29, 9, Yes, Frobenius_on_prime_field)."},"skills":["Frobenius Fixes Every Element of This Prime Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0086","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3db9f832bf3869350c2276e91ac2ab884e44a1a0cc0ebbc0bcdb8bfcfc318f3c","family_id":"qf-mod31-fielddeep-frobenius","seed":202610316086,"semantic_template":"qf-mod31-fielddeep-frobenius/{\"element\":10,\"p\":31}"},"fields":{"title":"Frobenius Fixes Every Element of This Prime Field","stem":"Use the complete prime field Z_31, with prime p=31, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. element=10, possibly zero. Give (element,p,power,equal?,Frobenius_on_prime_field), with power=element^p modulo p.","answer":{"kind":"expression","value":"(10, 31, 10, Yes, Frobenius_on_prime_field)","explanation":"Compute the p-th power in the specified prime field. This includes zero and does not assert that x^p=x holds in every finite extension field. The exact result, retaining the requested order, is (10, 31, 10, Yes, Frobenius_on_prime_field)."},"skills":["Frobenius Fixes Every Element of This Prime Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0087","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"21a04baa3cdc4047405b3599f00a8695ad8c2ff8388ff9444cc9425a515a0d03","family_id":"qf-mod31-fielddeep-frobenius","seed":202610316087,"semantic_template":"qf-mod31-fielddeep-frobenius/{\"element\":11,\"p\":37}"},"fields":{"title":"Frobenius Fixes Every Element of This Prime Field","stem":"Use the complete prime field Z_37, with prime p=37, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. element=11, possibly zero. Give (element,p,power,equal?,Frobenius_on_prime_field), with power=element^p modulo p.","answer":{"kind":"expression","value":"(11, 37, 11, Yes, Frobenius_on_prime_field)","explanation":"Compute the p-th power in the specified prime field. This includes zero and does not assert that x^p=x holds in every finite extension field. The exact result, retaining the requested order, is (11, 37, 11, Yes, Frobenius_on_prime_field)."},"skills":["Frobenius Fixes Every Element of This Prime Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0088","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c02555b3e060779bedfd2db0bc328c27b80e4089813b6bb7f1d7f158bf0a2610","family_id":"qf-mod31-fielddeep-frobenius","seed":202610316088,"semantic_template":"qf-mod31-fielddeep-frobenius/{\"element\":12,\"p\":41}"},"fields":{"title":"Frobenius Fixes Every Element of This Prime Field","stem":"Use the complete prime field Z_41, with prime p=41, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. element=12, possibly zero. Give (element,p,power,equal?,Frobenius_on_prime_field), with power=element^p modulo p.","answer":{"kind":"expression","value":"(12, 41, 12, Yes, Frobenius_on_prime_field)","explanation":"Compute the p-th power in the specified prime field. This includes zero and does not assert that x^p=x holds in every finite extension field. The exact result, retaining the requested order, is (12, 41, 12, Yes, Frobenius_on_prime_field)."},"skills":["Frobenius Fixes Every Element of This Prime Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0089","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b09735d23a180907a6063ebada55f8df0efebb81907e6cd7a396898e95379bc","family_id":"qf-mod31-fielddeep-frobenius","seed":202610316089,"semantic_template":"qf-mod31-fielddeep-frobenius/{\"element\":13,\"p\":43}"},"fields":{"title":"Frobenius Fixes Every Element of This Prime Field","stem":"Use the complete prime field Z_43, with prime p=43, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. element=13, possibly zero. Give (element,p,power,equal?,Frobenius_on_prime_field), with power=element^p modulo p.","answer":{"kind":"expression","value":"(13, 43, 13, Yes, Frobenius_on_prime_field)","explanation":"Compute the p-th power in the specified prime field. This includes zero and does not assert that x^p=x holds in every finite extension field. The exact result, retaining the requested order, is (13, 43, 13, Yes, Frobenius_on_prime_field)."},"skills":["Frobenius Fixes Every Element of This Prime Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0090","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b7b45d18507edac8fbe86a34dce8ffa5542bf217e6f2ac28165b2c84567cef13","family_id":"qf-mod31-fielddeep-frobenius","seed":202610316090,"semantic_template":"qf-mod31-fielddeep-frobenius/{\"element\":14,\"p\":47}"},"fields":{"title":"Frobenius Fixes Every Element of This Prime Field","stem":"Use the complete prime field Z_47, with prime p=47, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. element=14, possibly zero. Give (element,p,power,equal?,Frobenius_on_prime_field), with power=element^p modulo p.","answer":{"kind":"expression","value":"(14, 47, 14, Yes, Frobenius_on_prime_field)","explanation":"Compute the p-th power in the specified prime field. This includes zero and does not assert that x^p=x holds in every finite extension field. The exact result, retaining the requested order, is (14, 47, 14, Yes, Frobenius_on_prime_field)."},"skills":["Frobenius Fixes Every Element of This Prime Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0091","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ae646504200501141e90fc1112994680913aa2d253be17ac0a1c7f25ed243b67","family_id":"qf-mod31-fielddeep-frobenius","seed":202610316091,"semantic_template":"qf-mod31-fielddeep-frobenius/{\"element\":15,\"p\":53}"},"fields":{"title":"Frobenius Fixes Every Element of This Prime Field","stem":"Use the complete prime field Z_53, with prime p=53, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. element=15, possibly zero. Give (element,p,power,equal?,Frobenius_on_prime_field), with power=element^p modulo p.","answer":{"kind":"expression","value":"(15, 53, 15, Yes, Frobenius_on_prime_field)","explanation":"Compute the p-th power in the specified prime field. This includes zero and does not assert that x^p=x holds in every finite extension field. The exact result, retaining the requested order, is (15, 53, 15, Yes, Frobenius_on_prime_field)."},"skills":["Frobenius Fixes Every Element of This Prime Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0092","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1669076424dc091babe589225cc583886fea2fde9201d92ef2352a2314d97ce4","family_id":"qf-mod31-fielddeep-frobenius","seed":202610316092,"semantic_template":"qf-mod31-fielddeep-frobenius/{\"element\":16,\"p\":59}"},"fields":{"title":"Frobenius Fixes Every Element of This Prime Field","stem":"Use the complete prime field Z_59, with prime p=59, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. element=16, possibly zero. Give (element,p,power,equal?,Frobenius_on_prime_field), with power=element^p modulo p.","answer":{"kind":"expression","value":"(16, 59, 16, Yes, Frobenius_on_prime_field)","explanation":"Compute the p-th power in the specified prime field. This includes zero and does not assert that x^p=x holds in every finite extension field. The exact result, retaining the requested order, is (16, 59, 16, Yes, Frobenius_on_prime_field)."},"skills":["Frobenius Fixes Every Element of This Prime Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0093","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"750bd5be6107332869f7a1ffa03ccd2ec37e76b1f8324ce9ed2051de0f3ae8e4","family_id":"qf-mod31-fielddeep-frobenius","seed":202610316093,"semantic_template":"qf-mod31-fielddeep-frobenius/{\"element\":17,\"p\":61}"},"fields":{"title":"Frobenius Fixes Every Element of This Prime Field","stem":"Use the complete prime field Z_61, with prime p=61, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. element=17, possibly zero. Give (element,p,power,equal?,Frobenius_on_prime_field), with power=element^p modulo p.","answer":{"kind":"expression","value":"(17, 61, 17, Yes, Frobenius_on_prime_field)","explanation":"Compute the p-th power in the specified prime field. This includes zero and does not assert that x^p=x holds in every finite extension field. The exact result, retaining the requested order, is (17, 61, 17, Yes, Frobenius_on_prime_field)."},"skills":["Frobenius Fixes Every Element of This Prime Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0094","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"31469bbeb20749325bafc18654540fc818c472f9df25f8a839c7dca22091901e","family_id":"qf-mod31-fielddeep-frobenius","seed":202610316094,"semantic_template":"qf-mod31-fielddeep-frobenius/{\"element\":18,\"p\":67}"},"fields":{"title":"Frobenius Fixes Every Element of This Prime Field","stem":"Use the complete prime field Z_67, with prime p=67, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. element=18, possibly zero. Give (element,p,power,equal?,Frobenius_on_prime_field), with power=element^p modulo p.","answer":{"kind":"expression","value":"(18, 67, 18, Yes, Frobenius_on_prime_field)","explanation":"Compute the p-th power in the specified prime field. This includes zero and does not assert that x^p=x holds in every finite extension field. The exact result, retaining the requested order, is (18, 67, 18, Yes, Frobenius_on_prime_field)."},"skills":["Frobenius Fixes Every Element of This Prime Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0095","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"edc684b490e869bff26def66b35a1b468b8577d33571b8e68aac6fb928cd7d3a","family_id":"qf-mod31-fielddeep-frobenius","seed":202610316095,"semantic_template":"qf-mod31-fielddeep-frobenius/{\"element\":19,\"p\":71}"},"fields":{"title":"Frobenius Fixes Every Element of This Prime Field","stem":"Use the complete prime field Z_71, with prime p=71, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. element=19, possibly zero. Give (element,p,power,equal?,Frobenius_on_prime_field), with power=element^p modulo p.","answer":{"kind":"expression","value":"(19, 71, 19, Yes, Frobenius_on_prime_field)","explanation":"Compute the p-th power in the specified prime field. This includes zero and does not assert that x^p=x holds in every finite extension field. The exact result, retaining the requested order, is (19, 71, 19, Yes, Frobenius_on_prime_field)."},"skills":["Frobenius Fixes Every Element of This Prime Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0096","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cfaddeee0a6eae3288d46b5e8c5201ee4adcc2f39f6113c7c8697141d9f1a05c","family_id":"qf-mod31-fielddeep-frobenius","seed":202610316096,"semantic_template":"qf-mod31-fielddeep-frobenius/{\"element\":20,\"p\":73}"},"fields":{"title":"Frobenius Fixes Every Element of This Prime Field","stem":"Use the complete prime field Z_73, with prime p=73, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. element=20, possibly zero. Give (element,p,power,equal?,Frobenius_on_prime_field), with power=element^p modulo p.","answer":{"kind":"expression","value":"(20, 73, 20, Yes, Frobenius_on_prime_field)","explanation":"Compute the p-th power in the specified prime field. This includes zero and does not assert that x^p=x holds in every finite extension field. The exact result, retaining the requested order, is (20, 73, 20, Yes, Frobenius_on_prime_field)."},"skills":["Frobenius Fixes Every Element of This Prime Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0097","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5b0af063d45fe13236b842b0a3a65e6cf2d9a71334b1bad57de5ffdbacb33770","family_id":"qf-mod31-fielddeep-frobenius","seed":202610316097,"semantic_template":"qf-mod31-fielddeep-frobenius/{\"element\":21,\"p\":79}"},"fields":{"title":"Frobenius Fixes Every Element of This Prime Field","stem":"Use the complete prime field Z_79, with prime p=79, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. element=21, possibly zero. Give (element,p,power,equal?,Frobenius_on_prime_field), with power=element^p modulo p.","answer":{"kind":"expression","value":"(21, 79, 21, Yes, Frobenius_on_prime_field)","explanation":"Compute the p-th power in the specified prime field. This includes zero and does not assert that x^p=x holds in every finite extension field. The exact result, retaining the requested order, is (21, 79, 21, Yes, Frobenius_on_prime_field)."},"skills":["Frobenius Fixes Every Element of This Prime Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0098","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cb4c73506b71f478984b0196c970a3ffe72dee7188690088c12051f6a2276667","family_id":"qf-mod31-fielddeep-frobenius","seed":202610316098,"semantic_template":"qf-mod31-fielddeep-frobenius/{\"element\":22,\"p\":83}"},"fields":{"title":"Frobenius Fixes Every Element of This Prime Field","stem":"Use the complete prime field Z_83, with prime p=83, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. element=22, possibly zero. Give (element,p,power,equal?,Frobenius_on_prime_field), with power=element^p modulo p.","answer":{"kind":"expression","value":"(22, 83, 22, Yes, Frobenius_on_prime_field)","explanation":"Compute the p-th power in the specified prime field. This includes zero and does not assert that x^p=x holds in every finite extension field. The exact result, retaining the requested order, is (22, 83, 22, Yes, Frobenius_on_prime_field)."},"skills":["Frobenius Fixes Every Element of This Prime Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0099","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ba4dec05005927403dc61ced905d7f300afb6fce404172706dd9c9b76fc294cd","family_id":"qf-mod31-fielddeep-frobenius","seed":202610316099,"semantic_template":"qf-mod31-fielddeep-frobenius/{\"element\":23,\"p\":89}"},"fields":{"title":"Frobenius Fixes Every Element of This Prime Field","stem":"Use the complete prime field Z_89, with prime p=89, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. element=23, possibly zero. Give (element,p,power,equal?,Frobenius_on_prime_field), with power=element^p modulo p.","answer":{"kind":"expression","value":"(23, 89, 23, Yes, Frobenius_on_prime_field)","explanation":"Compute the p-th power in the specified prime field. This includes zero and does not assert that x^p=x holds in every finite extension field. The exact result, retaining the requested order, is (23, 89, 23, Yes, Frobenius_on_prime_field)."},"skills":["Frobenius Fixes Every Element of This Prime Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261031-r06-finite-field-0100","batch_id":"batch-20261014-advance-20261031-r06-finite-field-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"23d76fc1aefc56abb9bdf08a91eab6ed53694aa45d3fc301d71ec0afe8162946","family_id":"qf-mod31-fielddeep-frobenius","seed":202610316100,"semantic_template":"qf-mod31-fielddeep-frobenius/{\"element\":24,\"p\":97}"},"fields":{"title":"Frobenius Fixes Every Element of This Prime Field","stem":"Use the complete prime field Z_97, with prime p=97, residue labels 0 through p−1, and addition/multiplication modulo p. Nonzero inputs alone may be inverted; division by zero is never allowed. This prime-field family does not assert that every finite field has prime order or that every finite ring is a field. element=24, possibly zero. Give (element,p,power,equal?,Frobenius_on_prime_field), with power=element^p modulo p.","answer":{"kind":"expression","value":"(24, 97, 24, Yes, Frobenius_on_prime_field)","explanation":"Compute the p-th power in the specified prime field. This includes zero and does not assert that x^p=x holds in every finite extension field. The exact result, retaining the requested order, is (24, 97, 24, Yes, Frobenius_on_prime_field)."},"skills":["Frobenius Fixes Every Element of This Prime Field"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}