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).
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