On Classifying Extensions of $p$-adic Fields

TL;DR

This paper introduces practical methods to classify finite extensions of p-adic fields, especially cubic and tamely ramified extensions, based on polynomial coefficients, addressing complexities in wildly ramified cases.

Contribution

It provides explicit, closed-form techniques for identifying isomorphism classes of p-adic extensions using polynomial data, including handling wild ramification.

Findings

Method for classifying cubic extensions using polynomial coefficients

Approach for tamely ramified extensions of any prime degree

Discussion of complexities in wildly ramified cases

Abstract

Let $p$ be a prime and let $\mathbb{Q}_p$ be the field of $p$-adic numbers. It is known that the finite extensions of $\mathbb{Q}_p$ of a given degree are finite up to isomorphism. Given a cubic field extension $L$ of $\mathbb{Q}_p$ generated by the root of an irreducible polynomial $h$, we present a practical (closed-form) method to determine the isomorphism class in which $L$ lives, based on the coefficients of $h$. We discuss the subtleties of the wildly ramified case, when the degree of the extension coincides with $p$, the characteristic of the residue field. We also present a method for tamely ramified extensions of arbitrary prime degree.