I am looking for examples of CM fields whose Galois group is not abelian. By a CM field K I mean an imaginary quadratic extension of a totally real field $K\_0$. If the extension is not Galois I take the Galois closure $L/K$.


Obviously the degree of such a field is even.


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


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


For instance, if I consider $K=K\_0(\mu\_3)$, with $K\_0$ totally real of degree 4 and $\mu\_3$ the roots of unity of order 3, what are the possible Galois groups?


Thanks for your help!



Galois groups of CM fields
Let $K\_0$ be totally real of degree $2,3$ and $K/K\_0$ a totally imaginary quadratic extension. Let $L\_0$ and $L$ be the Galois closures of $K\_0$ and $K$ over ${\mathbb Q}$. Then $Gal(L)$ maps onto $Gal (L\_0)$ with kernel an abelian $2$ group (a vector space over ${\mathbb F}\_2$) . 


So the issue is: given a totally real number field $K\_0$ over ${\mathbb Q}$ (of degree $2,3$), what can its Galois group be? It is easy to see that it can be cyclic of order $2$, cyclic of order $3$ or $S\_3$. 


