Generalized cyclotomic periods

Abstract

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