Finite field

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

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 — The finite theorem provides inverses for all nonzero elements.
• Finite unital ring — Retain zero, one and both complete operation tables.
• 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.