# Finite field

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

A finite commutative unital ring is a finite field when every nonzero element has a multiplicative inverse.

mko-finite-field · definition · v0.10.0



## Mathematical statement

Use the complete finite ring tables to retain 1 distinct from 0, multiplicative commutativity and a two-sided inverse for every nonzero element. Division is defined only for nonzero denominators. A finite integral domain becomes a field through the finite injective-to-surjective multiplication-map theorem. A finite field has prime-power order, not necessarily prime order.





## Representative formula (exact TeX)

F(R)





## Assumptions


- The complete finite commutative unital-ring tables and actual zero and one are explicit.
- Inverses are checked for every nonzero element; division by zero is never allowed.





## Symbol scope


- F — finite field
- R — parent finite integral domain





## Intuition, definition, and misconceptions

Search completely for each nonzero element's inverse.

Keep the injective-to-surjective finite-theorem step from integral domain to field.

Finite-field order may be a power of a prime, not just a prime.

An arbitrary finite commutative ring is not a field; zero is not a denominator, and finite-field order need not be prime.





## Adopted derivation guide

[Certifying F3 and the finite-domain lemma](https://ocme.evemisslab.com/en/derivations/derivation-mko-finite-field-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Finite integral domain](https://ocme.evemisslab.com/en/mko/mko-finite-integral-domain/) — The finite theorem provides inverses for all nonzero elements.
- [Finite unital ring](https://ocme.evemisslab.com/en/mko/mko-finite-unital-ring/) — Retain zero, one and both complete operation tables.
- [Finite abelian group](https://ocme.evemisslab.com/en/mko/mko-finite-abelian-group/) — The additive-group structure remains complete.





## Verification boundary

Only the specified complete finite tables are classified; no arbitrary commutative-ring extension or omission of the nonzero inverse obligation is permitted.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.





## Next obligation

Add polynomials and extensions over finite fields.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/finite-field.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-finite-field/index.md)