On cubic equations over $P-$adic field

TL;DR

This paper establishes criteria for solving depressed cubic equations over p-adic fields, analyzes the applicability of the Cardano method, and generalizes known results from finite fields to p-adic contexts.

Contribution

It provides solvability criteria, counts solutions in various p-adic domains, and describes when the Cardano method applies to p-adic cubic equations.

Findings

Cardano method is not always applicable to p-adic cubics

Solutions counts are provided for different p-adic domains

Generalization of finite field results to p-adic fields

Abstract

We provide a solvability criteria for a depressed cubic equation in domains $\bz_p^{*},\bz_p,\bq_p$. We show that, in principal, the Cardano method is not always applicable for such equations. Moreover, the numbers of solutions of the depressed cubic equation in domains $\bz_p^{*},\bz_p,\bq_p$ are provided. Since $\bbf_p\subset\bq_p,$ we generalize J.-P. Serre's \cite{JPSJ} and Z.H.Sun's \cite{ZHS1,ZHS3} results concerning with depressed cubic equations over the finite field $\bbf_p$. Finally, all depressed cubic equations, for which the Cardano method could be applied, are described and the $p-$adic Cardano formula is provided for those cubic equations.