# Prime-Field Characteristic, Multiplicative Powers, Affine Equations and Frobenius

This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.

This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261014-advance-20261031-r06-finite-field-foundations · 2026-10-14 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.



## 1. Characteristic Is the Additive Order of One

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.

View answer and explanation(2, [0, 1, 0], 2, additive_order_of_one)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0001/)

Finite field · qf-mod31-fielddeep-characteristic · candidate





## 2. Characteristic Is the Additive Order of One

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.

View answer and explanation(3, [0, 1, 2, 0], 3, additive_order_of_one)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0002/)

Finite field · qf-mod31-fielddeep-characteristic · candidate





## 3. Characteristic Is the Additive Order of One

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.

View answer and explanation(5, [0, 1, 2, 3, 4, 0], 5, additive_order_of_one)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0003/)

Finite field · qf-mod31-fielddeep-characteristic · candidate





## 4. Characteristic Is the Additive Order of One

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.

View answer and explanation(7, [0, 1, 2, 3, 4, 5, 6, 0], 7, additive_order_of_one)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0004/)

Finite field · qf-mod31-fielddeep-characteristic · candidate





## 5. Characteristic Is the Additive Order of One

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.

View answer and explanation(11, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0], 11, additive_order_of_one)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0005/)

Finite field · qf-mod31-fielddeep-characteristic · candidate





## 6. Characteristic Is the Additive Order of One

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.

View answer and explanation(13, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0], 13, additive_order_of_one)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0006/)

Finite field · qf-mod31-fielddeep-characteristic · candidate





## 7. Characteristic Is the Additive Order of One

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.

View answer and explanation(17, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0], 17, additive_order_of_one)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0007/)

Finite field · qf-mod31-fielddeep-characteristic · candidate





## 8. Characteristic Is the Additive Order of One

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.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0008/)

Finite field · qf-mod31-fielddeep-characteristic · candidate





## 9. Characteristic Is the Additive Order of One

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.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0009/)

Finite field · qf-mod31-fielddeep-characteristic · candidate





## 10. Characteristic Is the Additive Order of One

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.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0010/)

Finite field · qf-mod31-fielddeep-characteristic · candidate





## 11. Characteristic Is the Additive Order of One

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.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0011/)

Finite field · qf-mod31-fielddeep-characteristic · candidate





## 12. Characteristic Is the Additive Order of One

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.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0012/)

Finite field · qf-mod31-fielddeep-characteristic · candidate





## 13. Characteristic Is the Additive Order of One

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.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0013/)

Finite field · qf-mod31-fielddeep-characteristic · candidate





## 14. Characteristic Is the Additive Order of One

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.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0014/)

Finite field · qf-mod31-fielddeep-characteristic · candidate





## 15. Characteristic Is the Additive Order of One

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.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0015/)

Finite field · qf-mod31-fielddeep-characteristic · candidate





## 16. Characteristic Is the Additive Order of One

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.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0016/)

Finite field · qf-mod31-fielddeep-characteristic · candidate





## 17. Characteristic Is the Additive Order of One

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.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0017/)

Finite field · qf-mod31-fielddeep-characteristic · candidate





## 18. Characteristic Is the Additive Order of One

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.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0018/)

Finite field · qf-mod31-fielddeep-characteristic · candidate





## 19. Characteristic Is the Additive Order of One

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.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0019/)

Finite field · qf-mod31-fielddeep-characteristic · candidate





## 20. Characteristic Is the Additive Order of One

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.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0020/)

Finite field · qf-mod31-fielddeep-characteristic · candidate





## 21. Characteristic Is the Additive Order of One

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.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0021/)

Finite field · qf-mod31-fielddeep-characteristic · candidate





## 22. Characteristic Is the Additive Order of One

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.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0022/)

Finite field · qf-mod31-fielddeep-characteristic · candidate





## 23. Characteristic Is the Additive Order of One

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.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0023/)

Finite field · qf-mod31-fielddeep-characteristic · candidate





## 24. Characteristic Is the Additive Order of One

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.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0024/)

Finite field · qf-mod31-fielddeep-characteristic · candidate





## 25. Characteristic Is the Additive Order of One

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.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0025/)

Finite field · qf-mod31-fielddeep-characteristic · candidate





## 26. The Nonzero Multiplicative Group Gives a p−1 Power Identity

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.

View answer and explanation(1, [1], 1, Yes, nonzero_multiplicative_group_cycle)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0026/)

Finite field · qf-mod31-fielddeep-power-cycle · candidate





## 27. The Nonzero Multiplicative Group Gives a p−1 Power Identity

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.

View answer and explanation(2, [2, 1], 1, Yes, nonzero_multiplicative_group_cycle)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0027/)

Finite field · qf-mod31-fielddeep-power-cycle · candidate





## 28. The Nonzero Multiplicative Group Gives a p−1 Power Identity

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.

View answer and explanation(3, [3, 4, 2, 1], 1, Yes, nonzero_multiplicative_group_cycle)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0028/)

Finite field · qf-mod31-fielddeep-power-cycle · candidate





## 29. The Nonzero Multiplicative Group Gives a p−1 Power Identity

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.

View answer and explanation(4, [4, 2, 1, 4, 2, 1], 1, Yes, nonzero_multiplicative_group_cycle)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0029/)

Finite field · qf-mod31-fielddeep-power-cycle · candidate





## 30. The Nonzero Multiplicative Group Gives a p−1 Power Identity

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.

View answer and explanation(5, [5, 3, 4, 9, 1, 5, 3, 4, 9, 1], 1, Yes, nonzero_multiplicative_group_cycle)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0030/)

Finite field · qf-mod31-fielddeep-power-cycle · candidate





## 31. The Nonzero Multiplicative Group Gives a p−1 Power Identity

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.

View answer and explanation(6, [6, 10, 8, 9, 2, 12, 7, 3, 5, 4, 11, 1], 1, Yes, nonzero_multiplicative_group_cycle)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0031/)

Finite field · qf-mod31-fielddeep-power-cycle · candidate





## 32. The Nonzero Multiplicative Group Gives a p−1 Power Identity

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.

View answer and explanation(7, [7, 15, 3, 4, 11, 9, 12, 16, 10, 2, 14, 13, 6, 8, 5, 1], 1, Yes, nonzero_multiplicative_group_cycle)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0032/)

Finite field · qf-mod31-fielddeep-power-cycle · candidate





## 33. The Nonzero Multiplicative Group Gives a p−1 Power Identity

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.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0033/)

Finite field · qf-mod31-fielddeep-power-cycle · candidate





## 34. The Nonzero Multiplicative Group Gives a p−1 Power Identity

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.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0034/)

Finite field · qf-mod31-fielddeep-power-cycle · candidate





## 35. The Nonzero Multiplicative Group Gives a p−1 Power Identity

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.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0035/)

Finite field · qf-mod31-fielddeep-power-cycle · candidate





## 36. The Nonzero Multiplicative Group Gives a p−1 Power Identity

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.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0036/)

Finite field · qf-mod31-fielddeep-power-cycle · candidate





## 37. The Nonzero Multiplicative Group Gives a p−1 Power Identity

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.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0037/)

Finite field · qf-mod31-fielddeep-power-cycle · candidate





## 38. The Nonzero Multiplicative Group Gives a p−1 Power Identity

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.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0038/)

Finite field · qf-mod31-fielddeep-power-cycle · candidate





## 39. The Nonzero Multiplicative Group Gives a p−1 Power Identity

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.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0039/)

Finite field · qf-mod31-fielddeep-power-cycle · candidate





## 40. The Nonzero Multiplicative Group Gives a p−1 Power Identity

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.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0040/)

Finite field · qf-mod31-fielddeep-power-cycle · candidate





## 41. The Nonzero Multiplicative Group Gives a p−1 Power Identity

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.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0041/)

Finite field · qf-mod31-fielddeep-power-cycle · candidate





## 42. The Nonzero Multiplicative Group Gives a p−1 Power Identity

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.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0042/)

Finite field · qf-mod31-fielddeep-power-cycle · candidate





## 43. The Nonzero Multiplicative Group Gives a p−1 Power Identity

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.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0043/)

Finite field · qf-mod31-fielddeep-power-cycle · candidate





## 44. The Nonzero Multiplicative Group Gives a p−1 Power Identity

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.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0044/)

Finite field · qf-mod31-fielddeep-power-cycle · candidate





## 45. The Nonzero Multiplicative Group Gives a p−1 Power Identity

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.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0045/)

Finite field · qf-mod31-fielddeep-power-cycle · candidate





## 46. The Nonzero Multiplicative Group Gives a p−1 Power Identity

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.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0046/)

Finite field · qf-mod31-fielddeep-power-cycle · candidate





## 47. The Nonzero Multiplicative Group Gives a p−1 Power Identity

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.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0047/)

Finite field · qf-mod31-fielddeep-power-cycle · candidate





## 48. The Nonzero Multiplicative Group Gives a p−1 Power Identity

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.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0048/)

Finite field · qf-mod31-fielddeep-power-cycle · candidate





## 49. The Nonzero Multiplicative Group Gives a p−1 Power Identity

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.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0049/)

Finite field · qf-mod31-fielddeep-power-cycle · candidate





## 50. The Nonzero Multiplicative Group Gives a p−1 Power Identity

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.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0050/)

Finite field · qf-mod31-fielddeep-power-cycle · candidate





## 51. Solve a Nonzero-Coefficient Affine Equation in the Prime Field

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?).

View answer and explanation(1, 1, 1, 1, 0, 1, Yes)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0051/)

Finite field · qf-mod31-fielddeep-affine-solve · candidate





## 52. Solve a Nonzero-Coefficient Affine Equation in the Prime Field

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?).

View answer and explanation(2, 2, 0, 2, 2, 0, Yes)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0052/)

Finite field · qf-mod31-fielddeep-affine-solve · candidate





## 53. Solve a Nonzero-Coefficient Affine Equation in the Prime Field

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?).

View answer and explanation(3, 3, 0, 2, 4, 0, Yes)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0053/)

Finite field · qf-mod31-fielddeep-affine-solve · candidate





## 54. Solve a Nonzero-Coefficient Affine Equation in the Prime Field

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?).

View answer and explanation(4, 4, 0, 2, 6, 0, Yes)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0054/)

Finite field · qf-mod31-fielddeep-affine-solve · candidate





## 55. Solve a Nonzero-Coefficient Affine Equation in the Prime Field

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?).

View answer and explanation(5, 5, 9, 9, 3, 9, Yes)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0055/)

Finite field · qf-mod31-fielddeep-affine-solve · candidate





## 56. Solve a Nonzero-Coefficient Affine Equation in the Prime Field

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?).

View answer and explanation(6, 6, 11, 11, 3, 11, Yes)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0056/)

Finite field · qf-mod31-fielddeep-affine-solve · candidate





## 57. Solve a Nonzero-Coefficient Affine Equation in the Prime Field

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?).

View answer and explanation(7, 7, 13, 5, 13, 13, Yes)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0057/)

Finite field · qf-mod31-fielddeep-affine-solve · candidate





## 58. Solve a Nonzero-Coefficient Affine Equation in the Prime Field

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?).

View answer and explanation(8, 8, 15, 12, 8, 15, Yes)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0058/)

Finite field · qf-mod31-fielddeep-affine-solve · candidate





## 59. Solve a Nonzero-Coefficient Affine Equation in the Prime Field

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?).

View answer and explanation(9, 9, 17, 18, 6, 17, Yes)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0059/)

Finite field · qf-mod31-fielddeep-affine-solve · candidate





## 60. Solve a Nonzero-Coefficient Affine Equation in the Prime Field

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?).

View answer and explanation(10, 10, 19, 3, 27, 19, Yes)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0060/)

Finite field · qf-mod31-fielddeep-affine-solve · candidate





## 61. Solve a Nonzero-Coefficient Affine Equation in the Prime Field

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?).

View answer and explanation(11, 11, 21, 17, 15, 21, Yes)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0061/)

Finite field · qf-mod31-fielddeep-affine-solve · candidate





## 62. Solve a Nonzero-Coefficient Affine Equation in the Prime Field

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?).

View answer and explanation(12, 12, 23, 34, 4, 23, Yes)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0062/)

Finite field · qf-mod31-fielddeep-affine-solve · candidate





## 63. Solve a Nonzero-Coefficient Affine Equation in the Prime Field

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?).

View answer and explanation(13, 13, 25, 19, 23, 25, Yes)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0063/)

Finite field · qf-mod31-fielddeep-affine-solve · candidate





## 64. Solve a Nonzero-Coefficient Affine Equation in the Prime Field

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?).

View answer and explanation(14, 14, 27, 40, 4, 27, Yes)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0064/)

Finite field · qf-mod31-fielddeep-affine-solve · candidate





## 65. Solve a Nonzero-Coefficient Affine Equation in the Prime Field

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?).

View answer and explanation(15, 15, 29, 22, 26, 29, Yes)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0065/)

Finite field · qf-mod31-fielddeep-affine-solve · candidate





## 66. Solve a Nonzero-Coefficient Affine Equation in the Prime Field

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?).

View answer and explanation(16, 16, 31, 10, 44, 31, Yes)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0066/)

Finite field · qf-mod31-fielddeep-affine-solve · candidate





## 67. Solve a Nonzero-Coefficient Affine Equation in the Prime Field

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?).

View answer and explanation(17, 17, 33, 7, 53, 33, Yes)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0067/)

Finite field · qf-mod31-fielddeep-affine-solve · candidate





## 68. Solve a Nonzero-Coefficient Affine Equation in the Prime Field

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?).

View answer and explanation(18, 18, 35, 17, 45, 35, Yes)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0068/)

Finite field · qf-mod31-fielddeep-affine-solve · candidate





## 69. Solve a Nonzero-Coefficient Affine Equation in the Prime Field

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?).

View answer and explanation(19, 19, 37, 60, 8, 37, Yes)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0069/)

Finite field · qf-mod31-fielddeep-affine-solve · candidate





## 70. Solve a Nonzero-Coefficient Affine Equation in the Prime Field

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?).

View answer and explanation(20, 20, 39, 32, 40, 39, Yes)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0070/)

Finite field · qf-mod31-fielddeep-affine-solve · candidate





## 71. Solve a Nonzero-Coefficient Affine Equation in the Prime Field

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?).

View answer and explanation(21, 21, 41, 7, 67, 41, Yes)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0071/)

Finite field · qf-mod31-fielddeep-affine-solve · candidate





## 72. Solve a Nonzero-Coefficient Affine Equation in the Prime Field

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?).

View answer and explanation(22, 22, 43, 18, 62, 43, Yes)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0072/)

Finite field · qf-mod31-fielddeep-affine-solve · candidate





## 73. Solve a Nonzero-Coefficient Affine Equation in the Prime Field

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?).

View answer and explanation(23, 23, 45, 65, 19, 45, Yes)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0073/)

Finite field · qf-mod31-fielddeep-affine-solve · candidate





## 74. Solve a Nonzero-Coefficient Affine Equation in the Prime Field

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?).

View answer and explanation(24, 24, 47, 26, 64, 47, Yes)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0074/)

Finite field · qf-mod31-fielddeep-affine-solve · candidate





## 75. Solve a Nonzero-Coefficient Affine Equation in the Prime Field

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?).

View answer and explanation(25, 25, 49, 66, 32, 49, Yes)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0075/)

Finite field · qf-mod31-fielddeep-affine-solve · candidate





## 76. Frobenius Fixes Every Element of This Prime Field

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.

View answer and explanation(0, 2, 0, Yes, Frobenius_on_prime_field)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0076/)

Finite field · qf-mod31-fielddeep-frobenius · candidate





## 77. Frobenius Fixes Every Element of This Prime Field

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.

View answer and explanation(1, 3, 1, Yes, Frobenius_on_prime_field)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0077/)

Finite field · qf-mod31-fielddeep-frobenius · candidate





## 78. Frobenius Fixes Every Element of This Prime Field

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.

View answer and explanation(2, 5, 2, Yes, Frobenius_on_prime_field)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0078/)

Finite field · qf-mod31-fielddeep-frobenius · candidate





## 79. Frobenius Fixes Every Element of This Prime Field

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.

View answer and explanation(3, 7, 3, Yes, Frobenius_on_prime_field)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0079/)

Finite field · qf-mod31-fielddeep-frobenius · candidate





## 80. Frobenius Fixes Every Element of This Prime Field

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.

View answer and explanation(4, 11, 4, Yes, Frobenius_on_prime_field)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0080/)

Finite field · qf-mod31-fielddeep-frobenius · candidate





## 81. Frobenius Fixes Every Element of This Prime Field

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.

View answer and explanation(5, 13, 5, Yes, Frobenius_on_prime_field)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0081/)

Finite field · qf-mod31-fielddeep-frobenius · candidate





## 82. Frobenius Fixes Every Element of This Prime Field

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.

View answer and explanation(6, 17, 6, Yes, Frobenius_on_prime_field)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0082/)

Finite field · qf-mod31-fielddeep-frobenius · candidate





## 83. Frobenius Fixes Every Element of This Prime Field

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.

View answer and explanation(7, 19, 7, Yes, Frobenius_on_prime_field)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0083/)

Finite field · qf-mod31-fielddeep-frobenius · candidate





## 84. Frobenius Fixes Every Element of This Prime Field

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.

View answer and explanation(8, 23, 8, Yes, Frobenius_on_prime_field)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0084/)

Finite field · qf-mod31-fielddeep-frobenius · candidate





## 85. Frobenius Fixes Every Element of This Prime Field

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.

View answer and explanation(9, 29, 9, Yes, Frobenius_on_prime_field)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0085/)

Finite field · qf-mod31-fielddeep-frobenius · candidate





## 86. Frobenius Fixes Every Element of This Prime Field

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.

View answer and explanation(10, 31, 10, Yes, Frobenius_on_prime_field)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0086/)

Finite field · qf-mod31-fielddeep-frobenius · candidate





## 87. Frobenius Fixes Every Element of This Prime Field

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.

View answer and explanation(11, 37, 11, Yes, Frobenius_on_prime_field)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0087/)

Finite field · qf-mod31-fielddeep-frobenius · candidate





## 88. Frobenius Fixes Every Element of This Prime Field

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.

View answer and explanation(12, 41, 12, Yes, Frobenius_on_prime_field)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0088/)

Finite field · qf-mod31-fielddeep-frobenius · candidate





## 89. Frobenius Fixes Every Element of This Prime Field

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.

View answer and explanation(13, 43, 13, Yes, Frobenius_on_prime_field)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0089/)

Finite field · qf-mod31-fielddeep-frobenius · candidate





## 90. Frobenius Fixes Every Element of This Prime Field

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.

View answer and explanation(14, 47, 14, Yes, Frobenius_on_prime_field)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0090/)

Finite field · qf-mod31-fielddeep-frobenius · candidate





## 91. Frobenius Fixes Every Element of This Prime Field

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.

View answer and explanation(15, 53, 15, Yes, Frobenius_on_prime_field)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0091/)

Finite field · qf-mod31-fielddeep-frobenius · candidate





## 92. Frobenius Fixes Every Element of This Prime Field

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.

View answer and explanation(16, 59, 16, Yes, Frobenius_on_prime_field)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0092/)

Finite field · qf-mod31-fielddeep-frobenius · candidate





## 93. Frobenius Fixes Every Element of This Prime Field

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.

View answer and explanation(17, 61, 17, Yes, Frobenius_on_prime_field)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0093/)

Finite field · qf-mod31-fielddeep-frobenius · candidate





## 94. Frobenius Fixes Every Element of This Prime Field

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.

View answer and explanation(18, 67, 18, Yes, Frobenius_on_prime_field)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0094/)

Finite field · qf-mod31-fielddeep-frobenius · candidate





## 95. Frobenius Fixes Every Element of This Prime Field

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.

View answer and explanation(19, 71, 19, Yes, Frobenius_on_prime_field)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0095/)

Finite field · qf-mod31-fielddeep-frobenius · candidate





## 96. Frobenius Fixes Every Element of This Prime Field

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.

View answer and explanation(20, 73, 20, Yes, Frobenius_on_prime_field)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0096/)

Finite field · qf-mod31-fielddeep-frobenius · candidate





## 97. Frobenius Fixes Every Element of This Prime Field

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.

View answer and explanation(21, 79, 21, Yes, Frobenius_on_prime_field)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0097/)

Finite field · qf-mod31-fielddeep-frobenius · candidate





## 98. Frobenius Fixes Every Element of This Prime Field

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.

View answer and explanation(22, 83, 22, Yes, Frobenius_on_prime_field)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0098/)

Finite field · qf-mod31-fielddeep-frobenius · candidate





## 99. Frobenius Fixes Every Element of This Prime Field

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.

View answer and explanation(23, 89, 23, Yes, Frobenius_on_prime_field)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0099/)

Finite field · qf-mod31-fielddeep-frobenius · candidate





## 100. Frobenius Fixes Every Element of This Prime Field

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.

View answer and explanation(24, 97, 24, Yes, Frobenius_on_prime_field)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r06-finite-field-0100/)

Finite field · qf-mod31-fielddeep-frobenius · candidate



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