On equation \(x^q=a\) over \(\mathbb Q_p\) (Q1762289)

Let \(\mathbb Q_p\) be the field of \(p\)-adic numbers, where \(p\) is a prime number. It is known that the equation \(x^2=a\), \(0\neq a=p^{\gamma(a)}(a_0+a_1p+a_2p^2+\dots)\), \(a_0>0\), has a solution \(x\in\mathbb Q_p\) if and only if (1) \(\gamma(a)\) is even; and (2) \(a_0\) is a quadratic residue modulo \(p\) if \(p\neq 2\); \(a_1=a_2=0\) if \(p=2\). In the paper the authors generalize this result for the equation \(x^q=a\) for any \(q\geq 2\).