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

Abstract

Let $p\ge 3$ 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 $n\ge 1$. 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 $m\ge 1$.

37 pages, accepted by The Ramanujan Journal

References in corpus (2)

• New identities for the partial Bell polynomials
• On binomial coefficients modulo squares of primes

Cited by in corpus (1)

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