Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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ON THE PRINCIPAL IDEAL THEOREM

  • Son, Jung-Je (DEPARTMENT OF MATHEMATICS KOREA UNIVERSITY) ;
  • Kwon, Soun-Hi (DEPARTMENT OF MATHEMATICS EDUCATION KOREA UNIVERSITY)
  • Published : 2007.07.30
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Abstract

In this paper we give an example of imaginary quadratic number field k such that every ideal of k becomes principal in some proper subfields of the Hilbert class field of k.

Keywords

References

  1. C. Batut, D. Bernardi, H. Cohen, and M. Olivier, PARI-GP, Universite de Bordeaux I, France. ftp://megrez.math.u-bordeaux.fr/pub/pari
  2. H. Cohen, A course in computational algebraic number theory, Graduate Texts in Mathematics, 138. Springer-Verlag, Berlin, 1993
  3. J. E. Cremona and R. W. K. Odoni, Some density results for negative Pell equations; an application of graph theory, J. London Math. Soc. (2) 39 (1989), no. 1, 16-28 https://doi.org/10.1112/jlms/s2-39.1.16
  4. F. Diaz y Diaz, On some families of imaginary quadratic fields, Math. Comp. 32 (1978), no. 142, 637-650 https://doi.org/10.2307/2006174
  5. G. Fujisaki, A note on a paper of Iwasawa, Proc. Japan Acad. Ser. A Math. Sci. 66 (1990), no. 2, 61-64 https://doi.org/10.3792/pjaa.66.61
  6. F. Gerth III, Some results on the capitulation problem for quadratic fields, Exposition. Math. 11 (1993), no. 2, 185-192
  7. G. Gras, Extensions abeliennes non ramifiees de degre premier d'un corps quadratique, Bull. Soc. Math. France 100 (1972), 177-193
  8. F.-P. Heider and B. Schmithals, Zur Kapitulation der Idealklassen in unverzweigten primzyklischen Erweiterungen, J. Reine Angew. Math. 336 (1982), 1-25
  9. K. Iwasawa, A note on capitulation problem for number fields, Proc. Japan Acad. Ser. A Math. Sci. 65 (1989), no. 2, 59-61 https://doi.org/10.3792/pjaa.65.59
  10. K. Iwasawa, A note on capitulation problem for number fields. II, Proc. Japan Acad. Ser. A Math. Sci. 65 (1989), no. 6, 183-186 https://doi.org/10.3792/pjaa.65.183
  11. K. Miyake, Algebraic investigations of Hilbert's Theorem 94, the principal ideal theorem and the capitulation problem, Exposition. Math. 7 (1989), no. 4, 289-346
  12. M. Muller, Berechnung Kapitulationskernen in quadratischen Zahlkorpen, Diplomarbeit, Mathematisches Institut II, Universitat Karlsruhe (TH), 1993
  13. J. Neukirch, Class field theory, Grundlehren der Mathematischen Wissenschaften, 280. Springer-Verlag, Berlin, 1986
  14. R. W. K. Odoni, A note on a recent paper of Iwasawa on the capitulation problem, Proc. Japan Acad. Ser. A Math. Sci. 65 (1989), no. 6, 180-182 https://doi.org/10.3792/pjaa.65.180
  15. A. Scholz and O. Taussky, Die Hauptideale der kubischen Klassenkorper imaginar-quadratischer Zahlkorper: ihre rechnerische Bestimmung und ihr EinfluB auf den Klassenkorperturm, J. Reine Angew. Math. 171 (1934), 19-41

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