Local and Global Residue Symbols for Algebraic Functions Fields

✍ Scribed by F. Schultheis

• Publisher: Elsevier Science
• Year: 1995
• Tongue: English
• Weight: 231 KB
• Volume: 52
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.1995.1059

✦ Synopsis

If (q) is a power of prime (p), we let (\mathrm{F}{4}) be a finite field with (q) elements, (R=\mathrm{F}{4}[x]) the polynomial ring over (\mathbb{F}{q}), and (k=\mathbb{F}{q}(x)) the rational function field. For any polynomial (M \in R). Carlitz [1] defined a "cyclotomic" extension (k_{M}) of (k). Let (K) be any finite, separable extension of (k_{M}). For certain (z, w \in K) with (w) and (M) relatively prime, we define an (M) th power residue symbol (the global symbol) ((z / w){K, M}). For any local field (E) that contains (k{M}) we define a local norm residue symbol ((\alpha, \beta); (E, M)_{\text {, }}) where (\beta) is the prime of (E) and (\alpha) and (\beta) are any elements of (E) with (\beta \neq 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). 1995 Academic Press. Inc.