(Theorem) Fundamental theorem of Galois theory

Overview

The fundamental theorem of Galois theory says that if a finite extension $E$ of a field $F$ is a Galois extension, then there is a bijection between the intermediate fields $K$ such that $E\le K\le F$ and the subgroups of the Galois group $\text{Gal}(E/F)$.

---

Theorem statement and proof

Fundamental theorem of Galois theory

Let $E$ be a Galois extension of a field $F$.

• (i) There is a bijective, order-reversing correspondence between subgroups of $\text{Gal}(E/F)$ and intermediate fields $F\le K\le E$, given as follows: every subgroup $H\le \text{Gal}(E/F)$ is associated to the fixed field $E^H$, and every intermediate field $F\le K\le E$ is associated to the subgroup $\text{Gal}(E/K)\le \text{Gal}(E/F)$: $\text{Gal}(E/E^H)=H,E^{\text{Gal}(E/K)}=K,$ where in particular, $E^{\text{Gal}(E/F)}=F$ and $E^e=E$. Further, since there are only finitely many subgroups of $\text{Gal}(E/F)$, there are only finitely many intermediate fields $K$ between $F$ and $E$.
• (ii) For every subgroup $H\le \text{Gal}(E/F)$, we have $[E:E^H]=\#(H)$, the number of elements in $H$, and hence $[E^H:F]=(\text{Gal}(E/F):H)=\#(\text{Gal}(E/F))/\#(H),$ the number of left cosets of $H$. Likewise, for every intermediate field $F\le K\le E$, $\#(\text{Gal}(E/K))=[E:K].$
• (iii) For every intermediate field $F\le K\le E$, the field $K$ is a normal extension of $F$ if and only if $\text{Gal}(E/K)\le \text{Gal}(E/F)$ is a normal subgroup. In this case, $K$ is a Galois extension of $F$ and $\text{Gal}(K/F)\cong \text{Gal}(E/F)/\text{Gal}(E/K).$

---

Code snippets

\text{Gal}(E/F)