Let $F\subset\mathbb{C}$ be a subfield and $f\in F[x]$. Let $K$ be the field generated by $F$ and the complex roots of $f$. Show that $K/L$ is Galois.

How do I show that? What is meant by complex roots? All roots (real numbers are in particular complex numbers) or just roots in $\mathbb{C}\setminus\mathbb{R}$?

For being Galois I have to show that the Extension is normal and separable. If we consider all roots of $f$, it is normal because it is the splitting field of $f$ right? How do I show separability?

  • $\begingroup$ A real number is also a complex number, so the complex roots are the roots in $\Bbb C$ $\endgroup$ – Kenny Lau Aug 9 '18 at 17:26

Separability is automatic for fields of characteristic $0$.

  • $\begingroup$ This does not provide an answer to the question. To critique or request clarification from an author, leave a comment below their post. - From Review $\endgroup$ – Adrian Keister Aug 9 '18 at 18:25
  • $\begingroup$ @AdrianKeister The question asks "How do I show separability?" which I perfectly answered. I do not understand what is wrong with this answer. $\endgroup$ – Kenny Lau Aug 9 '18 at 18:25
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    $\begingroup$ The OP asked for quite a few things in addition to separability. Generally, the length of your answer is more suited to a comment than an answer. $\endgroup$ – Adrian Keister Aug 9 '18 at 18:27

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