Galois Correspondence

For a finite Galois extension, intermediate fields correspond to subgroups of the Galois group.

Galois Correspondence

Theorem (Correspondence).
Let $L/K$ be a finite Galois extension and set $G=\mathrm{Gal}(L/K)$ . There is an inclusion-reversing bijection

H \le G \quad \longleftrightarrow \quad K \subseteq E \subseteq L

given by

H \mapsto L^H \quad \text{and} \quad E \mapsto \mathrm{Gal}(L/E),

where $L^H$ is the fixed field of $H$ .

This is the core bijection inside the fundamental theorem of Galois theory .

Examples.

If $G\cong C_2$ , then its only subgroups are $\{1\}$ and $G$ , so there are no nontrivial intermediate fields.
For $L=\mathbb{F}_{p^6}$ over $K=\mathbb{F}_p$ , subgroups of the cyclic group of order $6$ correspond to subfields $\mathbb{F}_{p^d}$ for $d\mid 6$ .
For $G=S_3$ , the subgroup lattice (order $1,2,3,6$ ) predicts the possible intermediate degrees in the splitting field of an irreducible cubic.

Related. Galois groups , degree/order formula .