On an explicit reciprocity law in local class field theory via $(\phi ,\Gamma )$-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 $n⩾2$. 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 $(\phi ,\Gamma )$-modules.

26 pages