Galois extension
Definition
Definition. An algebraic extension $K/k$ is called Galois iff it is separable and normal.
Definition. The group $\text{Aut}(K/k)$ of a Galois extension is called the Galois group. It is written as $\text{Gal}(K/k)$.
Properties
- If $k\subset K\subset\overline{k}$, then $\text{Gal}(K/k)\cong\sum_{K/k}^{\overline{k}/k}$.
- If $K/k$ is a finite Galois extension, then $#\text{Gal}(K/k)=[K:k]$.
- If $H\subset\text{Gal}(K/k)$ is a subgroup, then $$K^H\coloneqq{x\in K\mid\forall h\in H,\ h(x)=x}$$ is fixed and a field.
- Fundamental thm of Galois Theory