On a cyclotomic unit and related products (Q804628)

The author re-discovers part of Kummer's method for investigating pth powers in the pth cyclotomic field (that is, by looking at the logarithmic derivative of their \(\lambda\)-adic expansion), in proving that p divides \(\sum_{1\leq t\leq p/4}t^{p-3}\) whenever \(\prod_{1\leq k\leq p-1}(1+i\zeta^ k)^ k\) is a pth power in \({\mathbb{Z}}[i\zeta]\). Very similar criteria were given by Kummer in the case that the First Case of Fermat's Last Theorem is false for exponent p, which were recently generalized to the Second Case by \textit{F. Thaine} [J. Number Theory 29, 297-299 (1988; Zbl 0654.10018)].