CorollaryStatement: 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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Every finite extension of a finite field is simple

Statement

Every finite-degree extension $K / F$ of a finite field is simple: there exists $a \in K$ with $K = F (a)$ .

Facts & Assumptions

Given: A finite field $F$ and a finite extension $K / F$ of degree $n$ .

[L1]

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

[L2]

Degree $n$ gives a finite basis of $K$ over $F$ (The degree $[K : F] = \dim_{F} K$ of a finite field extension).

[L3]

The functions from an $n$ -element set to a finite set form a finite set (The set $A^{B}$ of functions $B \to A$ between finite sets is finite, with $∣ A^{B} ∣ = ∣ A ∣^{∣ B ∣}$ ).

[L4]

The subfield $F (a)$ is the smallest subfield containing $F$ and $a$ , and an extension equal to such a field is simple (Field extensions, generated subrings $F [S]$ , generated subfields $F (S)$ , and simple extensions).

Proof

technique · constructive

1.1

Coordinates in a finite basis identify $K$ with a finite set of functions from an $n$ -element index set to $F$ , so $K$ is a finite field.

givenL2L3

2.1

By [L1], choose a generator $a$ of the cyclic group $K^{\times}$ .

step 1.1L1chooseconstruct

3.1

The subfield $F (a)$ contains $0$ , $1$ , and every power of $a$ , hence all of $K^{\times}$ and therefore all of $K$ . Thus $K = F (a)$ by [L4].

step 2.1L4

4.1

The chosen $a$ exhibits the extension as simple.

step 3.1discharge-construct∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 87 results over 18 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 1 (standard reference, not scraped)