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Cyclotomy and combinatorial problems

Cyclotomy and combinatorial problems Dedicated to Professor H. HASSE on his seventieth birthday By S. CHOWLA w 1. The classical conjecture of GAVSS that the class-number of Q (V~g) is greater than 1 if i/ is a prime > 163, was settled only recently by H. ~r STARK [Proc. nat. Acad. Sc. U.S.A., 1967]. C.L. SIEGEL [Ge- sammelte Abhandlungen, Bd. 3, 440--42] has drawn attention to a similar unsolved conjecture, namely that the class-number of the cyclo- tomic field Q (exp [2r~i/g]) is greater then 1 if g (prime) ~ 23. In contrast to the situation of the imaginary quadratic field cited above, we do know in the cyclotomic case that hg ~ 1 for g :> go where go is a com- putable constant: This follows from the 1964 [GSttinger Nachrichten] paper of SIEGEL, as well as from an earlier paper of N. C. ANKENY and S. CHOWLA [Proo. Nat. Acztd. Sc. U.S.A., 1949]. ANKE~Y-C~OWLA and SIEGEL proved that log h* ~ g log, g where h* = h* (g) is the so-called first-factor of the class-number hg of the cyclotomic field Q(exp [2~i/g]). Hence ha > 1 for g > go. The problem suggested by SIEGEL has added interest in view http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg Springer Journals

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Publisher
Springer Journals
Copyright
Copyright
Subject
Mathematics; Mathematics, general; Algebra; Differential Geometry; Number Theory; Topology; Geometry
ISSN
0025-5858
eISSN
1865-8784
DOI
10.1007/BF02992803
Publisher site
See Article on Publisher Site

Abstract

Dedicated to Professor H. HASSE on his seventieth birthday By S. CHOWLA w 1. The classical conjecture of GAVSS that the class-number of Q (V~g) is greater than 1 if i/ is a prime > 163, was settled only recently by H. ~r STARK [Proc. nat. Acad. Sc. U.S.A., 1967]. C.L. SIEGEL [Ge- sammelte Abhandlungen, Bd. 3, 440--42] has drawn attention to a similar unsolved conjecture, namely that the class-number of the cyclo- tomic field Q (exp [2r~i/g]) is greater then 1 if g (prime) ~ 23. In contrast to the situation of the imaginary quadratic field cited above, we do know in the cyclotomic case that hg ~ 1 for g :> go where go is a com- putable constant: This follows from the 1964 [GSttinger Nachrichten] paper of SIEGEL, as well as from an earlier paper of N. C. ANKENY and S. CHOWLA [Proo. Nat. Acztd. Sc. U.S.A., 1949]. ANKE~Y-C~OWLA and SIEGEL proved that log h* ~ g log, g where h* = h* (g) is the so-called first-factor of the class-number hg of the cyclotomic field Q(exp [2~i/g]). Hence ha > 1 for g > go. The problem suggested by SIEGEL has added interest in view

Journal

Abhandlungen aus dem Mathematischen Seminar der Universität HamburgSpringer Journals

Published: Nov 17, 2008

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