Galois theory

Galois theory is an important tool for the study of fields. The primary objects of study in Galois theory are automorphisms of fields.

Consider the field $K=\mathbb{Q}(\sqrt{2})=\{a+b\sqrt{2}:a,b\in\mathbb{Q}\}$ . Then the map $f:K\to K$ given by $f(a+b\sqrt{2})=a-b\sqrt{2}$ is a field automorphism; that is, $f(\alpha\beta)=f(\alpha)f(\beta)$ and $f(\alpha+\beta)=f(\alpha)+f(\beta)$ , and $f$ is a bijection. Of course, the map $g:K\to K$ given by $g(\alpha)=\alpha$ is also a field automorphism. Both of these automorphisms are the identity automorphism on $\mathbb{Q}$ , a subfield of $K$ . It turns out that $f$ and $g$ are the only automorphisms of $K$ that fix $\mathbb{Q}$ . Furthermore, the automorphisms $f$ and $g$ form a group, called the Galois group of $K$ over $\mathbb{Q}$ .

We now define Galois groups more rigorously.

Let $L/K$ be a field extension. Then the set of field automorphisms of $L$ that fix $K$ form a group under composition. This group is called the Galois group of $L/K$ and is denoted $Gal(L/K)$ .

One may wonder if the elements of $K$ are the only elements of $L$ fixed by every element of $Gal(L/K)$ . It turns out that this is not always the case. For example, if $K=\mathbb{Q}$ and $L=\mathbb{Q}(\sqrt[3]{2})$ , then $Gal(L/K)$ is the trivial group, so every element of $L$ is fixed by $Gal(L/K)$ . If the elements of $K$ are the only elements of $L$ fixed by $Gal(L/K)$ , then we say that $L/K$ is a Galois extension.

Many beautiful results can be obtained with a bit of Galois theory. For example, one can prove that it is impossible to trisect an angle using Galois theory.

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Galois theory