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Sums of roots of unity in cyclotomic fields

Authors
Journal
Journal of Number Theory
0022-314X
Publisher
Elsevier
Publication Date
Volume
9
Issue
3
Identifiers
DOI: 10.1016/0022-314x(77)90067-1
Disciplines
  • Mathematics

Abstract

Abstract Let ζ n denote a primitive nth root of unity, n ≥ 4. For any integer k, 2 ≤ k ≤ n − 2 it is shown that the diophantine equation 1 + ζ n + ⋯ + ζ n k−1 = ϱα q has no solutions with ϱ, α in Q (ζ n ), ϱ a root of unity, α an algebraic integer, and q an integer ≥ 2, except when n = 10, 12, or 30, where the solutions are completely determined.

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