# Question 15136: Do Not Skip the Finite Injective-to-Surjective Step [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-0086) 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. R0029-15136 · q-20261014-advance-20261030-r01-finite-field-0086 ## 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, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] 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. Give (n,missing,field?,theorem_step,field_order_prime_power_not_necessarily_prime). theorem_step is finite_domain_injective_to_surjective_theorem or missing_nonzero_inverse. - G001 — Use the same already qualified complete finite commutative unital-ring tables on Fin(5)={0,…,4}, addition=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] 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. Give (n,missing,field?,theorem_step,field_order_prime_power_not_necessarily_prime). theorem_step is finite_domain_injective_to_surjective_theorem or missing_nonzero_inverse. ## 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, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]];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 finite commutative parent size=5;all missing nonzero inverse labels=[] After field qualification=Yes;finite theorem step=finite_domain_injective_to_surjective_theorem Conditions: none In a finite domain,nonzero cancellation gives injectivity;the finite same carrier makes it surjective,so one has a preimage/inverse.Missing inverses deny this table's field qualification,not the theorem.The finite step does not apply to arbitrary infinite domains. Step identifiers and dependenciesRule: finite_domain_field_theorem_not_definition_or_infinite_extension · Uses: G001, S006 S008 · impliesBefore field flag=Yes;finite carrier size=5 After field_order_prime_power_not_necessarily_prime Conditions: none For a qualified finite field,the characteristic is a prime p and finite vector-space dimension over the prime subfield gives size p^k.Do not restrict field order to primes or replace whole-table qualification by a prime-size shortcut. Step identifiers and dependenciesRule: finite_field_order_boundary_no_prime_size_shortcut · Uses: G001, S007 ## Reverse checks - S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 (5, [], Yes, finite_domain_injective_to_surjective_theorem, field_order_prime_power_not_necessarily_prime) 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: 4015811f7bd7eeff7c90f12c1cbedd5f50a4aa261e4487f0ff4b14dfbbc9beb8 - candidate record SHA-256: fea3848d70ef7292c64a81b16918b458b2a52687a1244afae5a89a1ff8333b2d - 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-0086.json)