On Hasse principle for \(x^n=a\) (Q1387933)

Let \(k\) be a, number field, let \(a\in k^*\) and let \(n>1\) be an integer. In this note it is proved that the equation \(x^n= a\) has a solution in \(k\) if and only if it has a solution in \(k_v\), for every place \(v\) of \(k\). This is called the Hasse principle for \(x^n= a\). As the authors remark, this is a special case of a more general theorem given by \textit{E. Artin} and \textit{J. Tate} [Class field theory, New York, Benjamin (1968; Zbl 0176.33504; see also Zbl 0681.12003)]. The proof given here uses properties of the Galois group of the splitting field of \(x^n=a\) and Chebotarev's density theorem. A consequence is that two elliptic curves \(E/k\) and \(E'/k\) are isomorphic over \(k\) if and only if they are isomorphic over every completion \(k_v\).