# Question 15158: Find a Two-Sided Inverse of the Specified Nonzero Element [Return to catalogue page 76](https://ocme.evemisslab.com/en/questions/page/76/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261030-r01-finite-field-foundations/#q-20261014-advance-20261030-r01-finite-field-0033) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0029-15158 · q-20261014-advance-20261030-r01-finite-field-0033 ## Original question and givens Use the same already qualified complete finite commutative unital-ring tables on Fin(5)={0,…,4}, addition=[[0, 1, 2, 3, 4], [1, 3, 4, 2, 0], [2, 4, 1, 0, 3], [3, 2, 0, 4, 1], [4, 0, 3, 1, 2]] and multiplication=[[0, 0, 0, 0, 0], [0, 3, 1, 4, 2], [0, 1, 2, 3, 4], [0, 4, 3, 2, 1], [0, 2, 4, 1, 3]] in carrier-label order. The actual zero and one are distinct and need not be labels 0 and 1. The inverse audit is additional to, not a replacement for, those ring axioms. Division by zero is never defined. A finite field has prime-power order, not necessarily prime order. element=3 is nonzero. Give (element,one,inverse,found?). - G001 — Use the same already qualified complete finite commutative unital-ring tables on Fin(5)={0,…,4}, addition=[[0, 1, 2, 3, 4], [1, 3, 4, 2, 0], [2, 4, 1, 0, 3], [3, 2, 0, 4, 1], [4, 0, 3, 1, 2]] and multiplication=[[0, 0, 0, 0, 0], [0, 3, 1, 4, 2], [0, 1, 2, 3, 4], [0, 4, 3, 2, 1], [0, 2, 4, 1, 3]] in carrier-label order. The actual zero and one are distinct and need not be labels 0 and 1. The inverse audit is additional to, not a replacement for, those ring axioms. Division by zero is never defined. A finite field has prime-power order, not necessarily prime order. element=3 is nonzero. Give (element,one,inverse,found?). ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-finite-field](https://ocme.evemisslab.com/en/mko/mko-finite-field/) ## Step-by-step derivation S001 · impliesBefore same complete carrier Fin(5);addition=[[0, 1, 2, 3, 4], [1, 3, 4, 2, 0], [2, 4, 1, 0, 3], [3, 2, 0, 4, 1], [4, 0, 3, 1, 2]];multiplication=[[0, 0, 0, 0, 0], [0, 3, 1, 4, 2], [0, 1, 2, 3, 4], [0, 4, 3, 2, 1], [0, 2, 4, 1, 3]] After actual zero=0;one=2;additive inverses=[0, 4, 3, 2, 1];additive commute/associate=[Yes, Yes];multiplicative commute/associate=[Yes, Yes];left/right distributive=[Yes, Yes] Conditions: none Even where the raw inverse oracle has looser guards,independently qualify the complete commutative unital parent declared by the source English scope,including every additive/distributive axiom and distinct identities.Do not skip them by name or inverse search alone. Step identifiers and dependenciesRule: field_stated_full_commutative_parent_not_raw_oracle_shortcut · Uses: G001 S002 · impliesBefore element label=0;actual zero=0;one=2;two-sided multiplicative inverse candidates=[] After selected inverse witness=null;nonzero inverse obligation=No Conditions: none Search the whole carrier for both products equaling the actual one.Nonzero is judged against the actual zero,not label0;zero has no inverse obligation. Step identifiers and dependenciesRule: field_all_nonzero_inverse_obligations_actual_labels · Uses: G001, S001 S003 · impliesBefore element label=1;actual zero=0;one=2;two-sided multiplicative inverse candidates=[4] After selected inverse witness=4;nonzero inverse obligation=Yes Conditions: none Search the whole carrier for both products equaling the actual one.Nonzero is judged against the actual zero,not label0;zero has no inverse obligation. Step identifiers and dependenciesRule: field_all_nonzero_inverse_obligations_actual_labels · Uses: G001, S002 S004 · impliesBefore element label=2;actual zero=0;one=2;two-sided multiplicative inverse candidates=[2] After selected inverse witness=2;nonzero inverse obligation=Yes Conditions: none Search the whole carrier for both products equaling the actual one.Nonzero is judged against the actual zero,not label0;zero has no inverse obligation. Step identifiers and dependenciesRule: field_all_nonzero_inverse_obligations_actual_labels · Uses: G001, S003 S005 · impliesBefore element label=3;actual zero=0;one=2;two-sided multiplicative inverse candidates=[3] After selected inverse witness=3;nonzero inverse obligation=Yes Conditions: none Search the whole carrier for both products equaling the actual one.Nonzero is judged against the actual zero,not label0;zero has no inverse obligation. Step identifiers and dependenciesRule: field_all_nonzero_inverse_obligations_actual_labels · Uses: G001, S004 S006 · impliesBefore element label=4;actual zero=0;one=2;two-sided multiplicative inverse candidates=[1] After selected inverse witness=1;nonzero inverse obligation=Yes Conditions: none Search the whole carrier for both products equaling the actual one.Nonzero is judged against the actual zero,not label0;zero has no inverse obligation. Step identifiers and dependenciesRule: field_all_nonzero_inverse_obligations_actual_labels · Uses: G001, S005 S007 · impliesBefore qualified field;selected nonzero element=3;actual one=2 After inverse=3;left product=2;right product=2 Conditions: none Existence follows from full field qualification;read back both products to one.The frozen inverse/division families contain no undefined output.Do not fabricate null for a missing inverse or invert zero. Step identifiers and dependenciesRule: selected_nonzero_field_inverse_not_zero_division · Uses: G001, S006 ## Reverse checks - S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result (3, 2, 3, Yes) The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 3c2c392a0085de8e5e8744ad7d507f6fd2b29c0717bcdb49d555b7c64caaa9e3 - candidate record SHA-256: 2d7cdb54f956e3e0c42ad603b79a2185105c3d81abfd362e0056b2f0e1b243ae - semantic review receipt SHA-256: 8c0ef6ca8c5ba2754e41f30c2b32bcb757756152d97421f7396d39038c11f725 - question-first witness SHA-256: c7920a0ff8e724b7e566ebbda30b31d8087c136974b62f8c760fbd26ab816117 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261030-r01-finite-field-0033.json)