Local Descriptions of Roots of Cubic Equations over P-adic Fields

Abstract

One of the most frequently asked question in the p-adic lattice models of statistical mechanics is that whether a root of a polynomial equation belongs to domains $${\mathbb {Z}}_p^{*}, {\mathbb {Z}}_p{\setminus }{\mathbb {Z}}_p^{*}, {\mathbb {Z}}_p, {\mathbb {Q}}_p{\setminus }{\mathbb {Z}}_p^{*}, {\mathbb {Q}}_p{\setminus }({\mathbb {Z}}_p{\setminus }{\mathbb {Z}}_p^{*}), {\mathbb {Q}}_p{\setminus }{\mathbb {Z}}_p, {\mathbb {Q}}_p $$ or not. However, this question was open even for lower-degree polynomial equations. In this paper, we give local descriptions of roots of cubic equations over the p-adic fields for $$p>3$$ .