Local and Global Residue Symbols for Algebraic Functions Fields Original Research Article

If q is a power of prime p, we let imageq be a finite field with q elements, R = imageq[x] the polynomial ring over imageq, and k = imageq(x) the rational function field. For any polynomial M set membership, variant R, Carlitz [1] defined a "cyclotomic" extension kM of k. Let K be any finite, separable extension of kM. For certain z, w set membership, variant 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) image where image 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).