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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
DOI: https://doi.org/10.1090/S0025-5718-1975-0409406-1
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?)

    G. GRAS, Sur les l-Classes d’Idéaux dans les Extensions Cycliques Relative de Degré Premier l, Thesis, Grenoble, 1972. C. S. HERZ, Construction of Class Fields, Seminar on Complex Multiplication, Lecture Notes in Math., vol. 21, Springer-Verlag, Berlin and New York, 1966.
  • Eizi Inaba, Ăśber 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
  • Daniel Shanks, The simplest cubic fields, Math. Comp. 28 (1974), 1137–1152. MR 352049, DOI https://doi.org/10.1090/S0025-5718-1974-0352049-8
  • W. ZINK, Thesis, Akademie der Wissenschaften der DDR, Berlin.

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Article copyright: © Copyright 1975 American Mathematical Society