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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.

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  • $\begingroup$ 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. $\endgroup$ – KCd Mar 25 at 23:58
  • $\begingroup$ @KCd Thanks for your comment. I edited the post. $\endgroup$ – rgb12 Mar 26 at 1:27
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You should look in books that discuss infinite Galois theory. Most general abstract algebra books don't have a section on that, so you won't find your question answered in such books.

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.

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  • $\begingroup$ I didn't know that I was looking references in wrong books. Thanks for pointing it out. Also thanks for answering first question. If you could help me on the second question, that would be great since I don't have any reference now. $\endgroup$ – rgb12 Mar 26 at 3:38

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