Separable Extensions and Finite Fields

Separability is the condition that distinguishes "well-behaved" extensions from pathological ones, and is automatic in characteristic zero. Finite fields provide a rich class of examples.

Separability

Definition7.5Separable polynomial and extension

A polynomial $f \in F[x]$ is separable if it has no repeated roots in $\overline{F}$ . Equivalently, $\gcd(f, f') = 1$ .

An algebraic element $\alpha$ is separable over $F$ if its minimal polynomial is separable. An algebraic extension $K/F$ is separable if every element of $K$ is separable over $F$ .

Theorem7.4Separability in characteristic zero

In characteristic zero (or more generally, over a perfect field), every algebraic extension is separable. Inseparability can occur only in characteristic $p > 0$ : the polynomial $x^p - a$ is inseparable iff $a \notin F^p$ .

ExampleAn inseparable extension

Let $F = \mathbb{F}_p(t)$ (rational functions over $\mathbb{F}_p$ ). The polynomial $x^p - t \in F[x]$ is irreducible (by Eisenstein with the prime $t$ ) and inseparable: in $\overline{F}$ , $x^p - t = (x - \alpha)^p$ where $\alpha^p = t$ . So $\alpha$ is inseparable over $F$ .

Finite Fields

Theorem7.5Classification of finite fields

For every prime $p$ and positive integer $n$ , there exists a field with $p^n$ elements, denoted $\mathbb{F}_{p^n}$ or $\mathrm{GF}(p^n)$ .
Any two fields with $p^n$ elements are isomorphic.
$\mathbb{F}_{p^n}$ is the splitting field of $x^{p^n} - x$ over $\mathbb{F}_p$ .
$\mathbb{F}_{p^m} \subseteq \mathbb{F}_{p^n}$ if and only if $m \mid n$ .

ExampleConstructing finite fields

$\mathbb{F}_4 = \mathbb{F}_2[x]/(x^2 + x + 1) = \{0, 1, \alpha, \alpha+1\}$ where $\alpha^2 + \alpha + 1 = 0$ .
$\mathbb{F}_8 = \mathbb{F}_2[x]/(x^3 + x + 1)$ .
$\mathbb{F}_9 = \mathbb{F}_3[x]/(x^2 + 1) = \{a + bi : a, b \in \mathbb{F}_3, i^2 = -1\}$ .

The Frobenius Automorphism

Definition7.6Frobenius automorphism

For a finite field $\mathbb{F}_{p^n}$ , the Frobenius automorphism is $\varphi: x \mapsto x^p$ . This is a field automorphism (in characteristic $p$ , $(a+b)^p = a^p + b^p$ ), and $\mathrm{Gal}(\mathbb{F}_{p^n}/\mathbb{F}_p) = \langle \varphi \rangle \cong \mathbb{Z}/n\mathbb{Z}$ .

RemarkThe Galois theory of finite fields

The extension $\mathbb{F}_{p^n}/\mathbb{F}_p$ is always Galois (normal and separable) with cyclic Galois group generated by the Frobenius. The subfields of $\mathbb{F}_{p^n}$ correspond bijectively to divisors of $n$ : for each $d \mid n$ , the unique subfield of order $p^d$ is $\mathrm{Fix}(\varphi^d)$ .