Three Properties of Galois Correspondence

The Fundamental Theorem of Galois Theory states that:

Given a field extension $E/F$ that is finite and Galois, there is a one-to-one correspondence between its intermediate fields and subgroups of its Galois group.
1) $H\leftrightarrow E^H$ where $H\leq\text{Gal}(E/F)$ and $E^H$ is the corresponding fixed field (the set of those elements in $E$ which are fixed by every automorphism in $H$).
2) $K\leftrightarrow\text{Aut}(E/K)$ where $K$ is an intermediate field of $E/F$ and $\text{Aut}(E/K)$ is the set of those automorphisms in $\text{Gal}(E/F)$ which fix every element of $K$.

This correspondence is a one-to-one correspondence if and only if $E/F$ is a Galois extension.

Three Properties of the Galois Correspondence

1. It is inclusing-reversing. The inclusion of subgroups $H_1\subseteq H_2$ holds iff the inclusion of fields $E^{H_2}\subseteq E^{H_1}$ holds.
2. If $H$ is a subgroup of $\text{Gal}(E/F)$, then $|H|=[E:E^H]$ and $|\text{Gal}(E/F)/H|=[E^H:F]$.
3. The field $E^H$ is a normal extension of $F$ (or equivalently, Galois extension, since any subextension of a separable extension is separable) iff $H$ is a normal subgroup of $\text{Gal}(E/F)$.

Author: mathtuition88

https://mathtuition88.com/

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