Use the same already qualified complete finite commutative unital-ring tables on Fin(5)={0,…,4}, addition=[[0, 1, 2, 3, 4], [1, 2, 4, 0, 3], [2, 4, 3, 1, 0], [3, 0, 1, 4, 2], [4, 3, 0, 2, 1]] and multiplication=[[0, 0, 0, 0, 0], [0, 4, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2, 3, 4, 1], [0, 3, 4, 1, 2]] 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, 4, 0, 3], [2, 4, 3, 1, 0], [3, 0, 1, 4, 2], [4, 3, 0, 2, 1]] and multiplication=[[0, 0, 0, 0, 0], [0, 4, 1, 2, 3], [0, 1, 2, 3, 4], [0, 2, 3, 4, 1], [0, 3, 4, 1, 2]] 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.
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.
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.
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.
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.
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.
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.
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.
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.
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.