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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?

Thanks for your help!

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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
    
But there is no reason for the extension to be Galois –  cmfield Mar 19 '13 at 7:54

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