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On 3-Class groups of certain pure cubic fields

Published online by Cambridge University Press:  17 April 2009

Frank Gerth III
Affiliation:
Mathematics Department, The University of Texas at Austin 1 University Station C1200, Austin, TX 78712–0257, United States of America, e-mail: [email protected]
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Recently Calegari and Emerton made a conjecture about the 3-class groups of certain pure cubic fields and their normal closures. This paper proves their conjecture and provides additional insight into the structure of the 3-class groups of pure cubic fields and their normal closures.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 2005

References

[1]Barrucand, P. and Cohn, H., ‘Remarks on principal factors in a relative cubic field,’ J. Number Theory 3 (1971), 226239.Google Scholar
[2]Barrucand, P., Williams, H. and Baniuk, L., ‘A computational technique for determining the class number of a pure cubic field’, Math. Comp. 30 (1976), 312323.Google Scholar
[3]Calegari, F. and Emerton, M., ‘On the ramification of Hecke algebras at Eisenstein primes’, Invent. Math. 160 (2005), 97144.CrossRefGoogle Scholar
[4]Gerth, F., ‘On 3-class groups of pure cubic fields’, J. Reine Angew. Math. 278/279 (1975), 5262.Google Scholar
[5]Gerth, F., ‘Ranks of 3-class groups of non-Galois cubic fields’, Acta Arith. 30 (1976), 307322.Google Scholar
[6]Gras, G., ‘Sur les l-classes d'idéaux des extensions non galoisiennes de ℚ de degré premier impair l a clôture galoisienne diédrale de degré 2l,’ J. Math. Soc. Japan 26 (1974), 677685.Google Scholar
[7]Honda, T., ‘Pure cubic fields whose class numbers are multiples of three,’ J. Number Theory 3 (1971), 712.CrossRefGoogle Scholar
[8]Washington, L., Introduction to cyclotonic fields (Springer-Verlag, New York, 1982).Google Scholar