On cubic equations over $P-$adic field

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.

18 pages

References in corpus (4)

• On equation $x^q=a$ over $\bq_p$
• Solvability criteria for the equation $x^q=a$ in the field of $p$-adic numbers
• Near NP-Completeness for Detecting p-adic Rational Roots in One Variable
• Faster p-adic Feasibility for Certain Multivariate Sparse Polynomials

Cited by in corpus (2)

• Phase transitions for $P$-adic Potts model on the Cayley tree of order three
• On equation $x^q=a$ over $\bq_p$