Galois Extension

Overview

Galois theory studies an algebraic field extension through its symmetries over the base field. Normality ensures that conjugate roots remain in the extension, and separability ensures that those roots are distinct. The two conditions do different work and are both necessary for the familiar correspondence between subgroups and intermediate fields.

Prerequisites

Use fields, algebraic extensions, minimal polynomials, and automorphisms. An automorphism over $K$ fixes every element of $K$. For a finite extension $L/K$, the degree $[L:K]$ is the dimension of $L$ as a $K$-vector space.

Definition and conventions

An algebraic extension $L/K$ is Galois when it is normal and separable. Normality means that every irreducible polynomial in $K[x]$ with a root in $L$ splits completely over $L$. Separability means that each element's minimal polynomial over $K$ has distinct roots in an algebraic closure.

Its Galois group is

For finite extensions, being Galois is equivalent to this group having exactly $[L:K]$ elements. The subgroup correspondence discussed below is stated only in this finite setting.

Intuition

Automorphisms can permute algebraic roots only while respecting every arithmetic relation. A Galois extension contains a complete separable collection of conjugates, so these permissible permutations capture the extension's structure.

Examples and non-examples

The extension $\mathbb Q(\sqrt2)/\mathbb Q$ is Galois. Its two automorphisms send $\sqrt2$ to $\sqrt2$ and to $-\sqrt2$. Their composition makes the Galois group cyclic of order two.

The extension $\mathbb Q(\sqrt[3]2)/\mathbb Q$ is separable but not normal: it lies in $\mathbb R$, while the other roots of $x^3-2$ are nonreal. Adjoining a primitive cube root of unity gives a splitting field.

In characteristic $p$, let $K=\mathbb F_p(t)$ and $L=K(u)$ with $u^p=t$. The polynomial $x^p-t$ is irreducible over $K$ and splits over $L$ as $(x-u)^p$. Thus $L/K$ is normal but inseparable, has degree $p$, and has only the identity $K$-automorphism. Normal extensions need not be separable, and normality never implies that a field is perfect.

Important properties

For finite Galois $L/K$, intermediate fields $E$ correspond to subgroups through

Inclusions reverse. The fixed-field theorem shows $[L:L^H]=|H|$, making the assignments inverse. An intermediate extension $E/K$ is Galois exactly when the corresponding subgroup is normal; then its Galois group is the quotient group.

Connections and further reading

Finite fields yield cyclic Galois groups generated by Frobenius. Algebraic number theory studies how these symmetries interact with prime ideals. See Milne, Chapters 2–3 for splitting, separability, and the finite correspondence.