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A note on $ 1$-class groups of number fields


Author: Frank Gerth
Journal: Math. Comp. 29 (1975), 1135-1137
MSC: Primary 12A35; Secondary 12A50, 12A30
MathSciNet review: 0409406
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Abstract: Let F be a number field and K a cyclic extension of degree l over F, where l is a rational prime. The l-class group of K is analyzed as a $ {\operatorname{Gal}}(K/F)$-module in the case where the l-class group of F is trivial. The resulting structure theorem is used to compute the structure of the 3-class groups of certain cyclic cubic fields that are discussed in a paper of D. Shanks.


References [Enhancements On Off] (What's this?)

  • [1] G. GRAS, Sur les l-Classes d'Idéaux dans les Extensions Cycliques Relative de Degré Premier l, Thesis, Grenoble, 1972.
  • [2] C. S. HERZ, Construction of Class Fields, Seminar on Complex Multiplication, Lecture Notes in Math., vol. 21, Springer-Verlag, Berlin and New York, 1966.
  • [3] Eizi Inaba, Über die Struktur der 𝑙-Klassengruppe zyklischer Zahlkörper vom Primzahlgrad 𝑙, J. Fac. Sci. Imp. Univ. Tokyo. Sect. I. 4 (1940), 61–115 (German). MR 0002999
  • [4] Daniel Shanks, The simplest cubic fields, Math. Comp. 28 (1974), 1137–1152. MR 0352049, 10.1090/S0025-5718-1974-0352049-8
  • [5] W. ZINK, Thesis, Akademie der Wissenschaften der DDR, Berlin.

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DOI: http://dx.doi.org/10.1090/S0025-5718-1975-0409406-1
Article copyright: © Copyright 1975 American Mathematical Society