I’m studying Galois theory, and have made my mind into seeing that there is a strong conection between a polynomial and it’s Galois group, but i’ve also seen that the Galois group is isomorphic to the group of automorphisms of the splitting field of the polynomial… so, if the theory is consistent, shouldn’t i be able to determine the Galois group of a given splitting field, without regarding the polynomial it came from? More specifically, my question is:
Given any finite simple field extension wich is also a splitting field, $\frac{\mathbb{F}(α)}{\mathbb{F}}$, can i always find a unique group of automorphisms isomorphic to the Galois Group of a Polynomial? If I can’t, what are the necessary and sufficient conditions for a field extension to be isomorphic to a Galois group?