The Fundamental Theorem of Galois Theory

The fundamental theorem establishes a bijective correspondence between intermediate fields of a Galois extension and subgroups of the Galois group, reversing inclusions.

Statement

Theorem8.3Fundamental theorem of Galois theory

Let $K/F$ be a finite Galois extension with Galois group $G = \mathrm{Gal}(K/F)$ . There is an inclusion-reversing bijection:

$\Phi: \{\text{intermediate fields } F \subseteq E \subseteq K\} \longleftrightarrow \{\text{subgroups } H \leq G\}$

given by $\Phi(E) = \mathrm{Gal}(K/E)$ and $\Phi^{-1}(H) = K^H$ . Moreover:

$[K:E] = |\mathrm{Gal}(K/E)|$ and $[E:F] = [G:\mathrm{Gal}(K/E)]$ .
$E/F$ is normal (hence Galois) if and only if $\mathrm{Gal}(K/E)$ is a normal subgroup of $G$ , and in this case $\mathrm{Gal}(E/F) \cong G/\mathrm{Gal}(K/E)$ .

Examples

ExampleGalois correspondence for $S_3$

$K = \mathbb{Q}(\sqrt[3]{2}, \omega)/\mathbb{Q}$ with $G \cong S_3$ . The subgroup lattice of $S_3$ has:

$\{e\}$ : fixed field $K$ .
$\langle (12) \rangle, \langle (13) \rangle, \langle (23) \rangle$ : three subgroups of order 2, fixing $\mathbb{Q}(\sqrt[3]{2}\omega^2)$ , $\mathbb{Q}(\sqrt[3]{2}\omega)$ , $\mathbb{Q}(\sqrt[3]{2})$ respectively.
$\langle (123) \rangle \cong \mathbb{Z}/3\mathbb{Z}$ : the unique normal subgroup of order 3, fixing $\mathbb{Q}(\omega)$ .
$S_3$ : fixed field $\mathbb{Q}$ .

Only $\mathbb{Q}(\omega)/\mathbb{Q}$ is Galois among the intermediate extensions (corresponding to the unique normal subgroup).

ExampleGalois correspondence for $\\mathbb{Z}/4\\mathbb{Z}$

$K = \mathbb{Q}(\zeta_5)/\mathbb{Q}$ with $G \cong (\mathbb{Z}/5\mathbb{Z})^\times \cong \mathbb{Z}/4\mathbb{Z} = \langle \sigma \rangle$ where $\sigma(\zeta_5) = \zeta_5^2$ . Subgroups: $\{e\}, \langle \sigma^2 \rangle, G$ . Intermediate field: $\mathbb{Q}(\zeta_5 + \zeta_5^4) = \mathbb{Q}(\sqrt{5})$ , fixed by $\sigma^2$ .

Normal Subgroups and Quotients

RemarkThe normal subgroup criterion

The correspondence between normal extensions and normal subgroups is one of the most powerful aspects: to determine if an intermediate field $E$ yields a Galois extension $E/F$ , one checks whether the corresponding subgroup is normal in $G$ . When it is, the quotient $G/H$ directly gives the Galois group of $E/F$ , avoiding the need to compute automorphisms directly.