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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

1.1

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 $d ∣ n$ .

givenL1L2

1.2

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 $t^{p^{n}} - t$ .

givenalgebra

2.1

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 $∣ F_{d} ∣ = p^{d}$ .

step 1.2L3L4

3.1

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 $K = F_{d}$ .

step 2.1L4

4.1

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.

step 1.1step 2.1step 3.1∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 90 results over 15 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, 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