Local and Global Residue Symbols for Algebraic Functions Fields

Abstract

If q is a power of prime p, we let Fq be a finite field with q elements, R = Fq[x] the polynomial ring over Fq, and k = Fq(x) the rational function field. For any polynomial M ∈ R, Carlitz [1] defined a "cyclotomic" extension kM of k. Let K be any finite, separable extension of kM. For certain z, w ∈ K with w and M relatively prime, we define an Mth power residue symbol (the global symbol) (z/w)K,M. For any local field E that contains kM we define a local norm residue symbol (α, β; E, M) p where p is the prime of E and α and β are any elements of E with β ≠ 0. Since these symbols are based on the additive theory of Carlitz′s cyclotomic function fields, these symbols are additive. We prove this result along with other basic properties of these symbols, including the equation that connects the two symbols (Theorem 16) and the continuity of the local symbol (Theorem 20).