Counting Extensions of $\mathfrak{p}$-Adic Fields with Given Invariants

TL;DR

This paper extends Krasner's mass formula to count $ extstyle ext{p}$-adic field extensions with specified invariants, including discriminant, ramification, and residual polynomial data, providing refined enumeration methods.

Contribution

It introduces two specialized formulas for counting $ extstyle ext{p}$-adic extensions with detailed invariants, enhancing previous enumeration techniques.

Findings

Derived formulas for counting extensions with given invariants.

Refined enumeration incorporating residual polynomial invariants.

Enhanced understanding of the structure of $ extstyle ext{p}$-adic extensions.

Abstract

We give two specializations of Krasner's mass formula. The first formula yields the number of extensions of a $\mathfrak{p}$-adic field with given, inertia degree, ramification index, discriminant, and ramification polygon. We then refine this formula further to the case where an additional invariant related to the residual polynomials of the segments of the ramification polygon is given.