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On Vanishing Sums of Roots of Unity

✍ Scribed by T.Y Lam; K.H Leung

Publisher
Elsevier Science
Year
2000
Tongue
English
Weight
150 KB
Volume
224
Category
Article
ISSN
0021-8693

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✦ Synopsis

An unsolved problem in number theory asked the following: For a given natural number m, what are the possible integers n for which there exist mth roots of unity

We show in this paper that the set of all possible n's is exactly the collection of -combinations of the prime divisors of m, where denotes the set of all non-negative integers. The proof is long and involves a subtle analysis of minimal vanishing sums of mth roots of unity, couched in the setting of integral group rings of finite cyclic groups. Our techniques also recovered with ease some of the classical results on vanishing sums of roots of unity, such as those of RΓ©dei, de Bruijn, and Schoenberg.

πŸ“œ SIMILAR VOLUMES

Vanishing Sums ofmth Roots of Unity in F
Vanishing Sums ofmth Roots of Unity in Finite Fields
✍ T.Y. Lam; K.H. Leung πŸ“‚ Article πŸ“… 1996 πŸ› Elsevier Science 🌐 English βš– 273 KB

In an earlier work, the authors determined all possible weights n for which there exists a vanishing sum 1 Ο© ΠΈ ΠΈ ΠΈ Ο© n Ο­ 0 of mth roots of unity i in characteristic 0. In this paper, the same problem is studied in finite fields of characteristic p. For given m and p, results are obtained on integers