Weil Reciprocity and Number Theory

Abstract

We begin with a treatment of classical algebraic number theory and algebraic $K$-theory. After introducing the notion of a (Steinberg) symbol, we use Tate's result on the structure of $K_2(\Q)$ to prove quadratic reciprocity. In a similar manner we give an explicit computation of $K_2(\Q(\sqrt{-2}))$ and derive an analogous reciprocity law. We then shift our focus to exploring the relationship between three reciprocity laws: Artin reciprocity, Weil reciprocity and quadratic reciprocity, and show how the global Artin map can be used to derive both quadratic and Weil reciprocity. Finally, we show how one can use Weil reciprocity to prove quadratic reciprocity as well.