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 polynomial equation with coefficients in a p-adic field Q_p belongs to domains Z_p^{*}, Z_p\Z_p^{*}, Q_p\Z_p, Q_p or not. This question is open even for lower degree polynomial equations. In this paper, we study this problem for a cubic equation of the general form. We provide solvability criteria and number of roots of the general cubic equation over the domains mentioned above.