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

TL;DR

This paper derives explicit formulas and recurrence relations for uniformizers in the false Tate curve extension of at

Contribution

The paper provides explicit formulas and recurrence polynomials for uniformizers in the false Tate curve extension, enabling systematic construction of these uniformizers.

Findings

Established explicit formulas for uniformizers in specific field extensions.

Proved existence of recurrence polynomials for general extensions.

Demonstrated systematic construction of uniformizers.

Abstract

In this article, we investigate the explicit formula for the uniformizers of the false-Tate curve extension of $\mathbb{Q}_p$. More precisely, we establish the formula for the fields ${\mathbb{K}}_p^{m,1}={\mathbb{Q}}_p(\zeta_{p^m}, p^{1/p})$ with $m\geq 1$ and for general $n\geq 2$, we prove the existence of the recurrence polynomials ${\mathcal{R}}_p^{m,n}$ for general field extensions ${\mathbb{K}}_p^{m, n}$ of ${\mathbb{Q}}_p$, which shows the possibility to construct the uniformizers systematically.