PropositionStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-16

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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A field with $q$ elements is the splitting field of $x^{q} - x$ over its prime subfield

Statement

If $F$ is a field with $q$ elements, then every $a \in F$ satisfies $a^{q} = a$ , and $F$ is the splitting field of $t^{q} - t$ over its prime subfield.

Facts & Assumptions

Given: A finite field $F$ of order $q$ .

[L1]

The group $F^{\times}$ is cyclic (The multiplicative group $F_{q}^{\times}$ of a finite field is cyclic).

[L3]

A splitting field is generated over the base by all roots of the polynomial (Polynomials that split and splitting fields of a polynomial or a family of polynomials).

[L4]

A nonzero degree- $q$ polynomial over a domain has at most $q$ distinct roots (A nonzero polynomial of degree $n$ over an integral domain has at most $n$ distinct roots).

Proof

technique · direct

1.1

The cyclic group $F^{\times}$ has order $q - 1$ , so every nonzero $a \in F$ satisfies $a^{q - 1} = 1$ and hence $a^{q} = a$ . The equality also holds for $a = 0$ .

givenL1algebra

2.1

Thus all $q$ elements of $F$ are roots of $t^{q} - t$ . By [L4] there are no other distinct roots in any extension, so the polynomial splits into its linear factors over $F$ .

step 1.1L4

3.1

The set of roots is all of $F$ , so it generates $F$ over its prime subfield. By [L3], $F$ is the splitting field.

step 2.1L3∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 42 results over 13 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

K. Conrad, Finite Fields, Section 2 (standard reference, not scraped)

A field with qq elements is the splitting field of xq−xxq−x over its prime subfield