Characterization of Galois Extensions

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).

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