Favourite Theorems: The Fundamental Theorem of Galois Theory

Favourite Math Theorems: The Fundamental Theorem of Galois Theory

Oct 10th, 2009

I spent many years as a full-time student of mathematics. One of my favourite theorems from my days as an undergraduate was the following from the field of abstract algebra:

The Fundamental Theorem of Galois Theory

If $L:K$ is a finite separable normal field extension of degree $n$ with Galois group $\Gamma$ . Then there is a bijection between the subfields $M$ of $L$ over $K$ and the subgroups $H$ of $\Gamma$ given by the correspondences

$\Psi: M \rightarrow \{\mbox{the elements of}\ \Gamma \ \mbox{fixing}\ M\}$
$\Phi: H \rightarrow \{\mbox{the fixed field of}\ H\}$

Under this correspondence,

$|\Gamma| = n$ ;
$\Phi\Psi = \iota_{fields}$ and $\Psi\Phi = \iota_{groups}$ ;
If $G_{1} \lneq G_{2} \leq \Gamma$ iff $\Phi(G_{1})\supsetneq \Phi(G_{2})$ ;
(That is, the correspondence is lattice inverting.)
$\Gamma(L/M^{\sigma}) = \Gamma(L/M)^{\sigma}$ ;
(That is, conjugate fields correspond to conjugate groups.)
If M is an intermediate field and $H \leq \Gamma$ fixes M then $[L:M]=|H|$
$[M:K]= |\Gamma|/|H|.$
An intermediate field $M$ is a normal extension of $K$ iff the subgroup $H$ of $\Gamma$ which fixes $M$ is a normal subgroup of $\Gamma$ iff for all $\sigma$ in $\Gamma$ , $M^{\sigma}=M$ ;
If an intermediate field $M$ is a normal extension of $K$ then $\Gamma(M/K)$ is isomorphic to $\Gamma/H$ , where $H$ is the subgroup of $\Gamma$ that fixes $M$ .