What is the Galois group of $x^n + (x-1)^n $?

Question

What is the Galois group of $x^n + (x-1)^n $ over the rationals in terms of the integer $n$ ? In case that is too hard , what is it for the first 20 integers ?

Answer 1

First of all notice that $f(x):=x^n+(x-1)^n=0$ is equivalent to $(\frac{x-1}{x})^n+1=0$. The roots $f$ are therefore of the form $x=1/(1-y)$ where $y$ are the roots of $y^n+1=0$, so the Galois group is the same as for the polynomial $x^n+1$.

I'm not sure what "Galois group of a polynomial" is when the polynomial is reducible, so let me determine the Galois group for every irreducible factor. For every odd divisor $k$ of $n$ there is one such a factor, its roots are the $2n/k$-th primitive roots of $1$. The Galois group is thus the multiplicative group $(\mathbb{Z}/(2n/k))^*$. (you might mean the product of these groups over all $k$'s, as that's the automorphism group of he ring $\mathbb{Q}[x]/(f)$)

edit (to take account of Qiaochu's comment): the splitting field of $f$ is $\mathbb{Q}(e^{\pi i/n})$, so the Galois group is $(\mathbb{Z}/(2n))^*$.