Local and global residue symbols for algebraic function fields (Q1892575)

The author defines a global \(M\)-th power residue symbol \((z/w)\) for any finite, separable extension \(K/k_M\) over the Carlitz \(M\)-th cyclotomic function field \(k_M = k (\Lambda_M)\), and defines a local norm residue symbol \((\alpha, \beta)_P\) for any local field \(E \supset k_M\). He then proves some basic properties of the two symbols analogous to the number field case, including the additivity, equation connecting them, and the continuity of the local one. The definitions are by \((z/Q) \equiv z^{(NQ - 1)/M} \pmod {QO_Q}\) and \((\alpha, \beta)_P = [\beta, E] (\gamma) - \gamma\) where \(M \in \mathbb{F}_q [x]\), \(k = \mathbb{F}_q (x)\), \(z,w \in K\), \(Q\) is a prime of \(K\), \(N\) the norm map from \(K\) to \(k\), \(O_Q\) the valuation ring of \(Q\), \(\alpha, \beta \text{(unit)} \in E\), \(P\) the prime of \(E\), \([\;,E]\) the local Artin map, and \(\gamma\) is algebraic over \(E\) with \(\gamma^M = \alpha\). The notations are from \textit{D. Goss} [J. Algebra 81, 107-149 (1983; Zbl 0516.12010)].