Galois Extension and Galois Group of an almost cyclotomic extension

Question

We have seen examples for $\mathbb{Q}[\zeta_n]/\mathbb{Q}$ which is commonly seen as a cyclotomic extension, but what does the structure of $\mathbb{Q}[\zeta_n^2]/\mathbb{Q}$ sturcture look like and what is its Galois group?

Answer 1

This is still a certain cyclotomic extension. If $n$ is odd, then $\zeta_n^2$ is equivalent to adjoining $\zeta_n,$ and otherwise it's equivalent to adjining $\zeta_{n/2}.$