# Galois group of CM fields

I am looking for examples of CM fields whose Galois group is not abelian. By a CM field $K$ I mean a totally imaginary quadratic extension of a totally real field $K_0$. If the extension is not Galois I take the Galois closure.

Obviously the degree of such a field is even.

When $[K: \mathbb{Q}]=2$, $K$ is an imaginary quadratic field, so its Galois group is $\mathbb{Z}/2$.

My question is: what are the possible Galois groups for $[K: \mathbb{Q}]=4$ and 6?

For instance, if I consider $K=K_0(\xi_3)$, with $K_0$ totally real of degree 4 and $\xi_3$ the roots of unity of order $3$, what are the possible Galois groups?

You have to open and close equation with the $sign in order for them to appear nice... – DonAntonio Mar 18 '13 at 18:03 Sorry for the mistake – cmfield Mar 18 '13 at 18:07 well for any galois extension$K$, with$[K/\mathbb{Q}]=4$the possible galois groups are$V4$or$\mathbb{Z}/2\$, so neither will be non-abelian. – Chris Birkbeck Mar 18 '13 at 23:57