Solve $1+X+X^2+...+X^n=0$.

Question

Is it possible to solve $$1+X+X^2+...+X^n=0,$$ by end? I'm asking that because I have to compute Galois group of its splitting field, and I don't see how to make progress. If $n+1$ is prime, I know that the Galois group is isomorphic to $\mathbb Z/p\mathbb Z$, but in general, I don't really know how to do this.

Answer 1

Hint

$$X^{n+1}-1=(X-1)(1+X+...+X^n).$$

Answer 2

This will be a cyclotomic extension since $(1-x)(1+x+\cdots+x^n)=1-x^{n+1}$

These are pretty well studied so you could try looking up that term