## Characterization of Galois Extensions

For a finite extension $E/F$, each of the following statements is equivalent to the statement that $E/F$ is Galois:

1) $E/F$ is a normal extension and a separable extension.
2) Every irreducible polynomial in $F[x]$ with at least one root in $E$ splits over $E$ and is separable.
3) $E$ is a splitting field of a separable polynomial with coefficients in $F$.
4) $|\text{Aut}(E/F)|=[E:F]$, that is, the number of automorphisms equals the degree of the extension.
5) $F$ is the fixed field of $\text{Aut}(E/F)$.

## Fundamental Theorem of Galois Theory

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.
$H\leftrightarrow E^H$

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 $H$).
$K\leftrightarrow\text{Aut}(E/K)$

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.

Examples
1) $E\leftrightarrow\{\text{id}_E\}$, the trivial subgroup of $\text{Gal}(E/F)$.
2) $F\leftrightarrow\text{Gal}(E/F)$.