Let $f(x)$ be an irreducible polynomial of degree $nd$ over $k$, and suppose there exists $g(x)\in k[x]$ (of degree $d$) such that $f(x)=g(x^n)$, i.e. all the terms of $f(x)$ have a degree which is a multiple of $n$. Can anything be said about the Galois group of $f(x)$ given this information? I suspect the answer may depend on whether $k$ contains $n$th roots of unity.
In the special case $d=n=2$, we have a biquadratic quartic, whose Galois group must be $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$, $\mathbb{Z}/4\mathbb{Z}$, or $D_8$. So, I was wondering if a result like this could be generalized.
