# If every irreducible polynomial in $K[x]$ is separable then every algebraic closure $\bar{K}$ of $K$ is Galois over $K$?

Let $$K$$ be a field.

1. If every irreducible polynomial in $$K[x]$$ is separable then every algebraic closure $$\bar{K}$$ of $$K$$ is Galois over $$K$$?

2. If every algebraic closure $$\bar{K}$$ of $$K$$ is Galois over $$K$$ then every algebraic extension of $$K$$ is separable over $$K$$?

Here is the list of related statements that I learned in class:

1. Let $$F$$ be an extension field of $$K$$ with $$\mathrm{char}K=p\neq 0$$. If $$u\in F$$ is algebraic over $$K$$, then $$u^{p^n}$$ is separable over $$K$$ for some $$n\geq 0$$.

2. If $$F$$ is an extension field of $$K$$, $$X$$ is a subset of $$F$$ such that $$F=K(X)$$, and every element of $$X$$ is separable over $$K$$, then $$F$$ is a separable extension of $$K.$$

3. If $$F$$ is a separable extension field of $$E$$ and $$E$$ is a separable extension field of $$K,$$ then $$F$$ is separable over $$K.$$

Definition: Let $$F$$ be an extension field of $$K$$ such that the fixed field of the Galois group $$\mathrm{Aut}_KF$$ is $$K$$ it self.

The statements in number 1 and 2 are the equivalent definitions of perfect field. I searched through algebra books that I have but couldn't find a reference. Any reference or ideas are greatly welcome.

• What have you tried already? Perhaps you can tell us what "Galois" means to you when describing an infinite-degree algebraic extension (like $\overline{K}/K$, usually) or what you know about intermediate extensions of separable extensions.
– KCd
Mar 25, 2020 at 23:58
• @KCd Thanks for your comment. I edited the post.
– user709182
Mar 26, 2020 at 1:27

For your first question, every element $$\alpha$$ of $$\overline{K}$$ has a separable minimal polynomial over $$K$$, so by Galois theory $$\alpha$$ lies in a finite Galois extension: the splitting field over $$K$$ of its minimal polynomial over $$K$$. Thus each finite extension of $$K$$ in $$\overline{K}$$ is contained in a finite Galois extension of $$K$$, so $$\overline{K}/K$$ is Galois.
Your definition makes no sense. You write "Let $$F$$ be an extension field of $$K$$ such that the fixed field of the Galois group $${\rm Aut}(F/K)$$ is $$K$$ itself." That is not a definition of anything.