On cubic equations over p-adic fields

Abstract

We provide a solvability criteria for a depressed cubic equation in domains Z * p , Zp, Qp. 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 Z * p , Zp, Qp are provided. Since Fp ⊂ Qp, we generalize J.-P. Serre's [27] and Z.H.Sun's [28,30] results concerning with depressed cubic equations over the finite field Fp. 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. Mathematics Subject Classification: 11Sxx

Solvability of Cubic Equations over Q3

“…One of methods to find solutions of the cubic equation in a local field is the Cardano method. However, by means of the Cardano method, we could not tell an existence of solutions of any cubic equations (Mukhamedov et al 2014.…”

“…Case I. We know (Mukhamedov et al 2014) that if the cubic equation (3) has a solution in then it is necessary to have either one of the following conditions: ⎜a⎜ 3 = ⎜b⎜ 3 ≥ 1 or ⎜b⎜ 3 < ⎜a⎜ 3 = 1 or ⎜a⎜ 3 < ⎜b⎜ 3 = 1. We shall study case by case.…”

“…This study is a continuation of papers Mukhamedov et al (2014, and Saburov and Ahmad (2014) where a solvability criterion for a cubic equation over the p-adic field  p , where p ≠ 3, was provided. In this paper, we shall provide a solvability criterion for the cubic equation over domains, ,  3 , and…”

“…The solvability criterion for the cubic equation over  p , for all prime p ≠ 3, was provided in papers Mukhamedov et al (2014 and Saburov and Ahmad (2014). This problem was open for the case p = 3 and we are aiming to solve it in this paper.…”

“…in  p ) but have solutions in  p (resp. in  p ) (Mukhamedov et al (2014. Therefore, finding a solvability criterion for the depressed cubic equation (2) in domains ,  p ,  p is of independent interest.…”