On an explicit reciprocity law in local class field theory via $(\varphi, \Gamma)$-modules

Abstract

Let $K$ be an unramified extension of $\mathbb{Q}_2$ and $\mu_{2^n}$ the group of $2^n$-th root of unity for a fixed integer $n\geqslant 2$. In this paper, we give an explicit formula for the $\mu_{2^n}$-valued Hilbert symbol over $K_n := K(\mu_{2^n})$ using the theory of $(\varphi, \Gamma)$-modules.