Galois Theory

Symmetries of fields; subgroups subfields.

The idea

Galois theory studies a field extension $K \supseteq F$ through its symmetries.

Definition (Galois group).

The Galois group $\operatorname{Gal}(K/F)$ of a field extension $K \supseteq F$ is the group, under composition, of field automorphisms of $K$ that fix every element of $F$. A finite extension is Galois when $|\operatorname{Gal}(K/F)| = [K : F]$.

An extension is Galois, then, when it has as many symmetries as its degree permits.

Theorem (Fundamental theorem of Galois theory).

Let $K/F$ be a finite Galois extension and let $G = \operatorname{Gal}(K/F)$. Then the intermediate fields $F \subseteq E \subseteq K$ correspond one-to-one with the subgroups of $G$: a subgroup $H$ corresponds to the field of elements fixed by every automorphism in $H$, and a field $E$ corresponds to the subgroup of automorphisms fixing $E$. The correspondence reverses inclusion.

The larger the subgroup, the smaller its fixed field: at the ends, the trivial subgroup $\{e\}$ corresponds to $K$ itself, and the whole group $G$ to the base field $F$.

The theorem turns questions about fields into questions about the subgroups of a single finite group: to list the intermediate fields of $K/F$ and see how they nest, we list the subgroups of $G$, a finite computation, and reverse their lattice.

Ways to work on it

• Walkthrough. The Galois group, its order, and the correspondence.
• Practice. Order of the Galois group from the degree.
• Hardest. Compute the Galois group of a degree-four extension, and see what the theory says about the quintic.

Not sure where to start? Take the ten-question placement test.