Explicit uniformizers for certain totally ramified extensions of the field of $p$-adic numbers

TL;DR

This paper constructs explicit uniformizers for a specific class of totally ramified extensions of the p-adic numbers, involving roots of unity and p-th roots, enhancing understanding of their structure.

Contribution

The paper provides explicit uniformizers for the extension it(3;2,1;p) of it(3;2) involving roots of unity and p-th roots, which was not previously explicitly known.

Findings

Explicit uniformizers constructed for the extension

Enhanced understanding of ramified extensions of it(3;2)

Potential applications in local field theory

Abstract

Let $p$ be an odd prime number. We construct explicit uniformizers for the totally ramified extension $\mathbb{Q}_p(\zeta_{p^2},\sqrt[p]{p})$ of the field of $p$-adic numbers $\mathbb{Q}_p$, where $\zeta_{p^2}$ is a primitive $p^2$-th root of unity.