# Galois extension in $\mathbb{C}$

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?

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

Separability is automatic for fields of characteristic $0$.