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

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 ${\mathbb{K}}_p^{m,1}={\mathbb{Q}}_p({\zeta }_{p^m},p^{1\mathrm{/}p})$ with $m\ge 1$ and for general $n\ge 2$, 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.