# Ramification groups of Galois extensions over local fields of positive characteristic with Galois group isomorphic to the group of unitriangular matrices

## TL;DR

This paper investigates the ramification groups in Galois extensions over local fields of positive characteristic with Galois groups isomorphic to unitriangular matrix groups, providing explicit formulas for ramification breaks.

## Contribution

It introduces a method to compute ramification groups directly from matrix valuations without using elements of the extension field.

## Key findings

- Upper ramification breaks are linear functions of matrix entry valuations.
- Explicit formulas for ramification groups are derived.
- Method simplifies ramification analysis in these Galois extensions.

## Abstract

We study the ramification groups of finite Galois extensions $L/K$ of a complete discrete valuation field $K$ of equal characteristic $p>0$ with perfect residue field and Galois group isomorphic to the group of unitriangular matrices $UT_n(\mathbb{F}_p)$ over $\mathbb{F}_p$. We show that the upper ramification breaks can be expressed as a linear function of the valuation of the entries of a matrix directly constructed from the coefficients of a defining equation of the extension. This allows us to compute the ramification groups without using any elements of $L-K$.