The congruence for Gauss period
Let \(K\) be a subfield of the cyclotomic field \(L= \mathbb{Q}(\zeta_ p)\), where \(\zeta_ p\) is a primitive \(p\)-th root of unity (\(p\) an odd prime). Let \(\pi\) denote a prime element in \(K\) dividing \(p\). The author studies the \(\pi\)-adic expansion of the Gaussian period \(\beta= \text{Tr}_{L/K} (\zeta_ p)\). He shows that if \(\pi\) is suitably chosen then \[ \beta\equiv k\sum_{i=0}^ n (1/ (ki)!) \pi^ i\pmod {\pi^{n+1}}, \] where \(k= [L:K]\) and \(n=[K: \mathbb{Q}]\). There are three applications concerning the Wieferich primes, the representation of \(4p\) by a quadratic form, and the number of solutions of a linear congruence \(\text{mod } p\).
scientific article; zbMATH DE number 638543