TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-16

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The subfields of $F_{p^{n}}$ are the unique fields $F_{p^{d}}$ for positive divisors $d$ of $n$

Statement

Let $F$ be a field of order $p^{n}$ . For each positive divisor $d$ of $n$ , $F$ has exactly one subfield of order $p^{d}$ , namely

$F_{d} = {a \in F : a^{p^{d}} = a} .$

These are all the subfields of $F$ .

Facts & Assumptions

Given: A finite field $F$ of order $p^{n}$ .

[L1]

Finite degrees multiply in a tower (Tower law for finite extensions: $[L : F] = [L : K] [K : F]$ ).

[L2]

A finite field has prime-power order, and its exponent is its degree over the prime field (Every finite field has order $p^{n}$ for a unique prime characteristic $p$ and positive integer $n$ ).

[L3]

The roots of $t^{p^{d}} - t$ form a subfield and are simple (In characteristic $p$ , the roots of $x^{p^{n}} - x$ form a subfield and are all simple).

[L4]

A field of order $p^{r}$ is the full root set and splitting field of $t^{p^{r}} - t$ (A field with $q$ elements is the splitting field of $x^{q} - x$ over its prime subfield).

Proof

technique · direct

1.1givenL1L2

If $K \subseteq F$ is a subfield, [L2] gives $∣ K ∣ = p^{d}$ and degrees $[K : F_{p}] = d$ , $[F : F_{p}] = n$ . The tower law [L1] gives $d ∣ n$ .

1.2givenalgebra

Now let $d ∣ n$ and write $n = e d$ . In characteristic $p$ , put $P_{r} (t) = t^{p^{r d}} - t$ . The identity $P_{r + 1} = P_{r}^{p^{d}} + (t^{p^{d}} - t)$ shows inductively that $t^{p^{d}} - t$ divides $t^{p^{n}} - t$ .

2.1step 1.2L3L4

By [L4], $t^{p^{n}} - t$ splits in $F$ . Hence its degree- $p^{d}$ divisor from step 1.2 splits there too. By [L3], its roots are distinct and form the subfield $F_{d}$ , so $∣ F_{d} ∣ = p^{d}$ .

3.1step 2.1L4

If $K \subseteq F$ has order $p^{d}$ , every element of $K$ satisfies $a^{p^{d}} = a$ by [L4], so $K \subseteq F_{d}$ . Both sets have $p^{d}$ elements, hence $K = F_{d}$ .

4.1step 1.1step 2.1step 3.1∎

Steps 1.1 and 2.1 give existence exactly for positive divisors $d$ of $n$ , and step 3.1 gives uniqueness and exhausts all subfields.

Depends on

Used by

Dependency tree · two levels

20 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

K. Conrad, Finite Fields, Theorem 2.8 (standard reference, not scraped)
J. S. Milne, Fields and Galois Theory, Propositions 4.19-4.24 (standard reference, not scraped)

The subfields of FpnFpn​ are the unique fields FpdFpd​ for positive divisors dd of nn