On cubic equations over \(p\)-adic fields (Q2876598)

For a prime number \(p>3\), let \(\mathbb{Q}_p\) denote the field of \(p\)-adic numbers and let \(\mathbb{Z}_p\) denote the ring of integers in \(\mathbb{Q}_p\). The authors provide a criterion for solvability in \(\mathbb{Z}_p^{\times}\), \(\mathbb{Z}_p\) and \(\mathbb{Q}_p\) of any cubic equation over \(\mathbb{Q}_p\). Moreover, the numbers of solutions of them are respectively calculated. Finally, the cubic equations over \(\mathbb{Q}_p\) for which the Cardano method is applicable, are determined.