Let \(L/k\) be a finite field extension. \(L/k\) is a Galois extension if it is normal and separable. In this case the Galois group of \(L/k\) is \[ \Gal(L/k) \coloneqq \Aut_{\Fieldsover{k}} (L). \]
Let \(L/k\) be a finite field extension, and let \([L:k]_s\) be its separable degree. The following are equivalent:
\(L/k\) is normal and separable.
The fixed field of \(\Aut_{\Fieldsover{k}}(L)\) is \(k\).
\(L\) is a splitting field over \(k\) of a separable polynomial \(p\in k[x]\).
\(L\) is a splitting field over \(k\) of an irreducible separable polynomial \(p\in k[x]\).
\(\size \Aut_{\Fieldsover{k}} (L) = [L: k]_s = [L: k]\).
The separable degree counts the embeddings of \(L\) into an algebraic closure that extend a fixed embedding of \(k\):
