Non cyclic Galois extension of degree 12

Question

Are there examples of Galois extensions of degree 12 which are not cyclic(i.e the corresponding Galois group is non cyclic)? If yes, please give me an hint on how to come up with such an extension?

Answer 1

Hint 1: Consider the polynomial $(x^2+1)(x^2-3)(x^3-2)$.

Hint 2: Show that the Galois group has two distinct elements of order $2$.

Answer 2

I would say that the easiest is just to adjoin the sixth root of some innocent prime like $7$, and make the extension normal by adjoining the sixth roots of unity, which is only a quadratic extension of the preceding. Thus $\Bbb Q(\sqrt{-3},\sqrt[6]7\,)$. Galois group is the dihedral group of order twelve.