# Proof of caracterization of Galois Extensions by Normals and Separable Extensions.

Theorem. Let $K$ be an extension of $F$. Then $K/F$ is Galois if only if $K/F$ is normal and separable.

In the proof that I know, the author add an intermediary statement for show that $$normal and separable$"$ implies $$Galois$"$, that is: $K$ is a splitting field of a set of separable polynomials over $F$.

This makes proof painful. However, I don't know if he adds this statement in order to improve the proof, or just to give one more criterion for characterization of Galois Extensions.

Anyway, I was wondering if there is proof (for converse) that doesn't use this statement. By the way, the reference I use is Morandi's book $$Field and Galois Theory$"$.

EDIT (due to Lubin's comment):

Let $K$ be an algebraic extension of $F$.

Definition 1. $K$ is Galois over $F$ is $F = \mathcal{F}(\mathrm{Gal}(K/F))$ where $\mathcal{F}$ represents the fixed field.

Definition 2. $K$ is normal over $F$ if $K$ is a splitting field of a set of polynomials over $F$.

Definition 3. $K$ is separable over $F$ if every $\alpha \in K$ is separable over $F$, i.e, $\min(F,\alpha)$ is separable over $F$.

• It depends on the definitions. What’s your definition of Galois? – Lubin Jun 22 '18 at 2:06
• @Lubin, good point! I'll write. – Corrêa Jun 22 '18 at 2:15