Fundamental thm of Galois Theory

Last updated January 27, 2022

Finite case

Suppose $K/k$ is a finite Galois extension. Let $G$ be its Galois group. Then

1. There exists a one-to-one correspondence \begin{gather} {\text{subgroups } H\subset G} \ \leftrightarrow \ {\text{subfields } k\subset M\subset K} \ H\mapsto K^H \ \text{Gal}(K/M)\leftarrow M \end{gather}
2. $M/k$ is normal iff $H\triangleleft G$ (normal subgroup)

Infinite case

Let $K/F$ be Galois, $G$ its Galois group equipped with the Krull topology.

1. There is an inclusion-reversing bijection \begin{gather}{\text{subextensions }K/L/F}\leftrightarrow{\text{closed subgroups }H\subset G} \ L\mapsto \text{Gal}(K/L) \ K^H \leftarrow H \end{gather}.
2. $L/F$ is Galois iff $\text{Gal}(K/L)\subset G$ is normal and open.
   1. $\text{Gal}(L/F)\cong G/\text{Gal}(K/L)$