A reciprocity law in function fields (Q6564139)

The author generalizes Gauss's Lemma to general function fields and then proves a reciprocity law for totally imaginary function fields; the value of \((\alpha/\beta)_n (\beta/\alpha)_n^{-1}\) is given as a power of \(-1\) depending only on the norms of \(\alpha\) and \(\beta\), and an \(n\)-th root of unity depending on the leading coefficients of \(\phi_\alpha\) and \(\phi_\beta\) for a suitably defined rank one Drinfeld module \(\phi\).