Galois Theory, 5th Edition

Chapter 12The Galois Correspondence

DOI: 10.1201/9781003213949-12

We are at last in a position to establish the fundamental properties of the Galois correspondence between a field extension and its Galois group. Most of the work has already been done, and all that remains is to put the pieces together.

12.1 The Fundamental Theorem of Galois Theory

Recall a few points of notation from Chapter 8. Let $L/K$ be a field extension in $ℂ$ with Galois group G, which consists of all K-automorphisms of L. Let $\mathcal{F}$ be the set of intermediate fields, that is, subfields M such that $K\subseteq M\subseteq L$, and let $\mathcal{G}$ be the set of all subgroups H of G. We defined two maps

$$\begin{array}{l}{*}^{}:\mathcal{F}\to \mathcal{G}{}^{†}:\mathcal{G}\to \mathcal{F}\hfill \end{array}$$

as follows: if $M\in \mathcal{F}$, then M^* is the group of all M-automorphisms of L. If $H\in \mathcal{G}$, then H^† is the ...