The study on general cubic equations over p-adic fields

Abstract

A Diophantine problem means to find all solutions of an equation or system of equations in integers, rational numbers, or sometimes more general number rings. The most frequently asked question is whether a root of a polynomial equation with coefficients in a p-adic field ℚp belongs to domains ℤ∗p, ℤp \ ℤ∗p, ℚp \ ℤp, ℚp or not. This question is open even for lower degree polynomial equations. In this paper, this problem is studied for cubic equations in a general form. The solvability criteria and the number of roots of the general cubic equation over the mentioned domains are provided.