Splitting field of $x^{n}-1$ over $\mathbb{Q}$

From I.N.Herstein's Topics in Algebra. Chap 5 Sec 5.3 Page 227 Problem 8

• Problem 8: If $n>1$ prove that the splitting field of $x^{n}-1$ over the field of rational numbers is of degree $\Phi(n)$ where $\Phi$ is the Euler $\Phi$-function. ( This is a well known theorem. I know of no easy solution, so don't be disappointed if you fail to get it. If you get an easy proof, I would like to see it.)

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.

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It's in section 13.6 in Dummit & Foote. – Bruno Stonek Feb 27 '11 at 19:12
@Bruno: Yeah i got it. – anonymous Feb 28 '11 at 4:50

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 $n$.