Relative norms of Gauss sums
Consider the Gaussian sum \(S(\kappa)=\sum \kappa (u)e^{2\pi i tr(u)/p}\), where \(\kappa (u)\equiv u^{-(N{\mathfrak p}-1)/m}\) (mod \({\mathfrak p})\) is a fixed m-th power residue character modulo \({\mathfrak p}\), \({\mathfrak p}\) is an unramified prime in the m-th cyclotomic field \({\mathbb{Q}}(\zeta)\) lying above the odd rational prime p of residue degree f, \(tr(u)=u+u^ p+...+u^{p^{f-1}}\) and \(\zeta =e^{2\pi i/m}\). Now S(\(\kappa)\) lies in K(\(\zeta)\) for some abelian field K of conductor p and the author is interested in an exact calculation of \(N_{K(\zeta)/Kk}S(\kappa)\) where k is a subfield of \({\mathbb{Q}}(\zeta)\). He achieves this when m is odd and greater than 3 and k is a quadratic field, by making good use of Stickelberger's theorem and the analytic class number formula. The result is a nice application of this style of artihmetic.
scientific article; zbMATH DE number 3983308