Galois Group of $x^{12}+x^{11}+\dots+x^2+x+1$

Question

It seems I have forgot all the details about the group theory. Anyone knows what is the Galois Group of $x^{12}+x^{11}+\dots+x^2+x+1$ and is that solvable?

Thanks.

Answer 1

Hints:

$$x^{12} + \ldots + x + 1 = \frac{x^{13} - 1}{x - 1}$$

The galois groups of $\,x^p-1\;$ over the rationals is isomorphic with $\,\left(\Bbb Z/p\Bbb Z\right)^*\;$ and thus has order equal to $\,\phi(p)=p-1\;$