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I would like an explanation for the fact stated in the title. To repeat:

Question: How does one prove that if a field has a separable extension of degree $n$, then it has a Galois extension of degree $n$?

Note that the statement might as well be false (though I would not bet on that). [EDIT: I thought I had a proof that the statement was true for $n=3$, but as Emil Jeřábek points out in his comments, this is false as well!]

This might be completely trivial, but I have no idea on how to address this question, and since it kinds of naturally arose in my research, I thought it would be appropriate to ask here.

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    $\begingroup$ That is not true. Let $K$ be the maximal solvable extension of $\mathbb{Q}$ (or $\mathbb{C}(t)$, etc.). There are irreducible quintic polynomials over $K$ (otherwise every quintic over $\mathbb{Q}$ would have a solvable Galois group). Thus, there are separable, degree $5$ extensions of $K$. However, every Galois extension of degree $5$ would have cyclic Galois group. Since $K$ is already the maximal solvable extension of $\mathbb{Q}$, there is no nontrivial, finite cyclic extension of $K$. $\endgroup$ – Jason Starr Sep 20 '18 at 12:00
  • $\begingroup$ So, you just confirmed my fear that the question is trivial. Thank you very much for your help, that was actually exactly what I was hoping for. $\endgroup$ – thierry stulemeijer Sep 20 '18 at 12:04
  • $\begingroup$ The reason it holds for $n=3$ is that every transitive subgroup of $S_n$ has a normal subgroup of index $n$, but this is false for $n\ge5$. $\endgroup$ – Emil Jeřábek Sep 20 '18 at 12:05
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    $\begingroup$ Sorry, you're right, it actually does not. But then I think your property fails already for $n=3$. Let $K/F$ be any non-normal extension of degree 3, and let $L/F$ be its normal closure. We have $[L:K]=2$. Using Zorn's lemma, let $F'$ be a maximal algebraic extension of $F$ such that $F'\cap L=F$, and let $K'$ and $L'$ be the composita of $F'$ with $K$ and $L$, respectively. We still have that $K'$ is a degree-3 extension of $F'$, and $L'$ is its normal closure over $F'$, of degree 2 over $K'$. $Gal(L'/F')\simeq S_3$. Assume for contradiction that $F'$ has a degree-3 Galois extension $H$... $\endgroup$ – Emil Jeřábek Sep 20 '18 at 14:32
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    $\begingroup$ ... By maximality of $F'$, $H$ has a nontrivial intersection with $L$, thus $H\cap L'$ is a proper extension of $F'$; since it is of prime degree, this implies $H\subseteq L'$. But then $Gal(L'/H)$ is a normal subgroup of $Gal(L'/F')$ of index 3, which as you observed, does not exist. $\endgroup$ – Emil Jeřábek Sep 20 '18 at 14:37
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I am posting my comment as an answer (I had no intention of causing any dispute).

That is not true. Let $K$ be the maximal solvable extension of $\mathbb{Q}$ (or $\mathbb{C}(t)$, etc.). There are irreducible quintic polynomials over $K$ (otherwise every quintic over $\mathbb{Q}$ would have a solvable Galois group). Thus, there are separable, degree 5 extensions of $K$. However, every Galois extension of degree 5 would have cyclic Galois group. Since $K$ is already the maximal solvable extension of $\mathbb{Q}$, there is no nontrivial, finite cyclic extension of $K$.

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  • $\begingroup$ That is probably the best course of action. I am thus deleting my own post. Thank you very much again for your help! $\endgroup$ – thierry stulemeijer Sep 20 '18 at 13:43

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