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Character coordinates and annihilators of cyclotomic numbers. (English) Zbl 0624.12006

“The object of this paper is a representation theoretical approach to the problem of determining all \({\mathbb{Q}}\)-linear relations between conjugate numbers in a cyclotomic field.” It is based on the ‘character coordinates’ of numbers in an abelian number field introduced by H.- W. Leopoldt [J. Reine Angew. Math. 201, 119-149 (1959; Zbl 0098.034)]. “We apply our method to relations between the numbers \(\cot^{(m)}(\pi k/n)\), \(\tan^{(m)}(\pi k/n)\), \(\cos ec^{(m)}(2\pi k/n)\), \(\sec^{(m)}(2\pi k/n)\), respectively, where m is \(\geq 0\) and \((k,n)=1\). Thereby we complete work of Chowla, Hasse, Jager-Lenstra, and others.”
Reviewer: H.Opolka

MSC:

11R18 Cyclotomic extensions
11J85 Algebraic independence; Gel’fond’s method

Citations:

Zbl 0098.034
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References:

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