Let \(n\geq 1\) and let \(\zeta_n\in\CC\) be a primitive \(n\)th root of unity. The cyclotomic field of \(n\)th roots of unity is \(\QQ(\zeta_n)\), the splitting field of \(x^n-1\) over \(\QQ\) [@DF04].
More generally, for a field \(k\) whose characteristic does not divide \(n\), the splitting field of \(x^n-1\) over \(k\) is \(k(\zeta)\) for any primitive \(n\)th root of unity \(\zeta\) in it.