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

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 Zp∗,Zp\Zp∗,Zp,Qp\Zp∗,Qp\(Zp\Zp∗),Qp\Zp,Qp 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