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


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


Zbl 0098.034
Full Text: DOI EuDML


[1] AYOUB, R., On a theorem of S.Chowla, J.Number Th.7, 105-107 (1975) · Zbl 0315.12004
[2] BAKER, A., BIRCH, B.J., and WIRSING, E. A., On a problem of Chowla, J.Number Th.5, 224-236 (1973) · Zbl 0267.10065
[3] BERNDT, B. C., and SCHOENFELD, L., Periodic analogues of the Euler-Maclaurin and Poisson summation formulas with applicatios in number theory, Acta Arith.28, 23-68 (1975) · Zbl 0268.10008
[4] CHOWLA, S., A special infinite series, Kong.Norsk. Vidensk.Selsk.Forhandl.37, 85-87 (1964) · Zbl 0223.10002
[5] CHOWLA, S., The nonexistence of nontrivial relations between the roots of a certain irreducible equation, J.Number Th.2, 120-123 (1970) · Zbl 0211.07005
[6] FLORIAN, A.,and PRACHAR, K., On the diophantine equation tan (k?/m)=k tan?/m, Monatsh.Math.102, 263-266 (1986) · Zbl 0597.10017
[7] GIRSTMAIR, K., Letter to the editor, J.Number Th.23, 405 (1986) · Zbl 0585.10021
[8] GIRSTMAIR, K., Ein v.Staudt-Clausenscher Satz f?r periodische Bernoullizahlen, Monatsh.Math.,to appear · Zbl 0626.12001
[9] HASSE, H., Vorlesungen ?ber Zahlentheorie, Berlin-G?ttingen-Heidelberg: Springer 1950
[10] HASSE, H., On a question of S.Chowla, Acta Arith.18, 275-280 (1971)
[11] JAGER, H.,and LENSTRA jr.,H.W., Linear independence of cosecant values, Nieuw Archief Wisk.(3)23, 131-144 (1975) · Zbl 0311.10017
[12] LEOPOLDT, H.W., Eine Verallgemeinerung der Bernoullischen Zahlen, Abh.Math.Sem.Hamburg22, 131-140 (1958) · Zbl 0080.03002
[13] LEOPOLDT, H. W., ?ber die Hauptordnung der ganzen Elemente eines abelschen Zahlk?rpers, J.reine angew. Math.201, 119-149 (1959) · Zbl 0098.03403
[14] OBERST, U., Anwendungen des chinesischen Restsatzes, Expo.Math.3, 97-148 (1985) · Zbl 0591.13002
[15] OKADA, T., On an extension of a theorem of S.Chowla, Acta Arith.38, 341-345 (1981)
[16] WANG, K., On a theorem of S.Chowla, J.Number Th.15, 1-4 (1982) · Zbl 0491.10024
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