Enumerating Extensions of $(\pi)$-Adic Fields with Given Invariants

TL;DR

This paper presents an algorithm to construct all extensions of a $()$-adic field with specified invariants such as inertia degree, ramification index, and discriminant, by generating minimal defining polynomials.

Contribution

It introduces a novel algorithm that enumerates all $()$-adic field extensions with given invariants through minimal polynomial construction.

Findings

Algorithm successfully constructs all extensions with specified invariants.

Provides a systematic method for enumerating $()$-adic field extensions.

Enhances understanding of the structure of $()$-adic fields with prescribed properties.

Abstract

We give an algorithm that constructs a minimal set of polynomials defining all extension of a $(\pi)$-adic field with given, inertia degree, ramification index, discriminant, ramification polygon, and residual polynomials of the segments of the ramification polygon.