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

Abstract

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 ${\mathbb{Z}}_p^{\ast },{\mathbb{Z}}_p\setminus {\mathbb{Z}}_p^{\ast },{\mathbb{Z}}_p,{\mathbb{Q}}_p\setminus {\mathbb{Z}}_p^{\ast },{\mathbb{Q}}_p\setminus \left({\mathbb{Z}}_p\setminus {\mathbb{Z}}_p^{\ast }\right),{\mathbb{Q}}_p\setminus {\mathbb{Z}}_p,{\mathbb{Q}}_p$ 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 $p>3$.

19 pages