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 $K$ .