Uniformizer of the False Tate Curve Extension of $\mathbb{Q}_p$

TL;DR

This paper provides explicit expansions of primitive roots of unity in a $p$-adic field and constructs a uniformizer for certain extensions of $Q_p$, advancing understanding of their algebraic structure.

Contribution

It offers an explicit formula for the expansion of roots of unity in a $p$-adic Mal'cev-Neumann field and constructs a uniformizer for specific $p$-adic extensions.

Findings

Explicit formula for the first countably infinite terms of the expansion of $oldsymbol{oldsymbol{ ext{ extit{p}}^n}$-th roots of unity.

Construction of a uniformizer for the extension $oldsymbol{K_{2,m}}$ of $oldsymbol{Q_p}$.

Enhanced understanding of the algebraic structure of $p$-adic extensions.

Abstract

Let $p\geq 3$ be a prime number. In this article, we study the canonical expansion of the primitive $p^n$-th root of unity $\zeta_{p^n}$ in $p$-adic Mal'cev-Neumann field $\mathbb{L}_p$ for $n\geq 1$. More precisely, we give the explicit formula for the first $\aleph_0$ terms of the expansion of $\zeta_{p^n}$ and as an application, we use it to construct a uniformizer of $K_{2,m}=\mathbb{Q}_p\left(\zeta_{p^2},p^{1/p^m}\right)$ with $m\geq 1$.