Generalized cyclotomic periods

Abstract:

Let $n$ and $q$ be relatively prime integers with $n > 1$, and set $N$ equal to twice the product of the distinct prime factors of $n$. Let $t(n)$ denote the order of $q$ $(\bmod n)$. Write $\eta = \sum _{\upsilon = 0}^{t(n) - 1}{a_\upsilon }\zeta _n^{{q^\upsilon }}$ where ${\zeta _n} = \exp (2\pi i/n)$. If ${a_\upsilon } = 1$ for all $\upsilon$, then $\eta$ is Kummer’s cyclotomic period, and if ${a_\upsilon } = \exp (2\pi i\upsilon /t(n))$ for each $\upsilon$, then $\eta$ is a type of Lagrange resolvent. For certain classes of ${a_\upsilon } \in {\mathbf {Q}}(\zeta _n^N)$, necessary and sufficient conditions for the vanishing of $\eta$ are given, and the degree of $\eta$ over ${\mathbf {Q}}$ is determined.