From I.N.Herstein’s Topics in Algebra. Chap 5 Sec 5.3 Page 227 Problem 8
First, I would like to see a proof of this result. Next, I think I have seen this proof in Dummit and Foote’s Abstract Algebra book, but not sure. Anyway, next question is: Has an easy solution been found to this problem? If not, I would like to know what efforts have been taken to make the proof more simple. And why does Herstein think an easy solution can exist.
I think the difficulty is proving that the $n$th cyclotomic polynomial is irreducible. Wikipedia says it’s a non-trivial result. This gives a factorization of $x^n-1$ as the product of all cyclotomic polynomials $\Phi_d$ for $d$ dividing $n$.