# Cyclotomic Fields and Cyclotomic Polynomials

In lectures we defined the cyclotomic polynomials as $\Phi_n(x):=\prod_{k=1,\gcd(k,n)=1}^n (x-e^{2\pi i k/n})$ and the cyclotomic field $K_n$ as the splitting field of $x^n-1$ over $\mathbb Q$. But we never explicitly talked about the relationship between the two.

Are the following assertions correct?

(1) Let $\xi$ be a primtive $n$-th root of unity, then $K_n=\mathbb Q(\xi)$.

(2) Let $\xi$ be a primtive $n$-th root of unity, then $\Phi_n(x)$ is its minimal polynimial over $\mathbb Q$.

(3) $K_n$ is the splitting field of $\Phi_n(x)$ over $\mathbb Q$.

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Yes, yes and yes. Any more or less decent book in Galois theory deals with this. – DonAntonio Mar 3 '13 at 19:32