Galois Theory

Question 1

What is the Galois group of extension E over F?

Answer 1

Gal(E/F) is the set of all automorphisms of E that fix F.

Question 2

Fixed field of subgroup, H, of Gal(E/F)?

Answer 2

E_H = {x in E | phi(x) = x for all phi in H}.

Question 3

Main idea of FTGT?

Answer 3

The number of subfields of E containing F is the number of subgroups of Gal(E/F).

Question 4

For any subfield K of E containing F, [E:K] = ?

[K:F] = ?

Answer 4

[E:K] = |Gal(E/K)|
[K:F] = |Gal(E/F)| / |Gal(E/K)|