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Remark on the class number of $ {\bf Q}(\sqrt{2p})$ modulo $ 8$ for $ p\equiv 5\;({\rm mod}\,8)$ a prime


Authors: Kenneth S. Williams and Christian Friesen
Journal: Proc. Amer. Math. Soc. 93 (1985), 198-200
MSC: Primary 11R11; Secondary 11R29
DOI: https://doi.org/10.1090/S0002-9939-1985-0770517-0
MathSciNet review: 770517
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Abstract: An explicit congruence modulo 8 is given for the class number of the real quadratic field $ Q(\sqrt {2p} )$, where $ p$ is a prime congruent to 5 modulo 8.


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

  • [1] Phillipe Barkan, Une propriété de congruence de la longueur de la période d'un développement en fraction continue, C. R. Acad. Sci. Paris Sér. A 281 (1975), 825-828.
  • [2] -, Sur des propriétés de congruence $ (\mod 4)$ liées à l'équation de Pell-Fermat, C. R. Acad. Sci. Paris Sér. A 289 (1979), 303-306.
  • [3] Bruce C. Berndt, Classical theorems on quadratic residues, L'Enseignement Math. 22 (1976), 261-304. MR 0441835 (56:229)
  • [4] Ezra Brown, Class numbers of real quadratic number fields, Trans. Amer. Math. Soc. 190 (1974), 99-107. MR 0364172 (51:427)
  • [5] P. G. L. Dirichlet, Recherches sur diverses applications de l'analyse infinitésimale à la théorie des nombres, J. Reine Angew. Math. 21 (1840), 134-155.
  • [6] Carl Friedrich Gauss, Letter to P. G. L. Dirichlet dated 30 May 1828. (Reproduced in P. G. L. Dirichlet's Werke, Vol. 2, Chelsea, New York, pp. 378-380.)
  • [7] J. W. L. Glaisher, On the expressions for the number of classes of a negative determinant, and on the numbers of positives in the octants of $ P$, Quart. J. Math. 34 (1903), 178-204.
  • [8] H. Holden, On various expressions for $ h$, the number of properly primitive classes for a negative determinant not containing a square factor (fifth paper), Messenger Math. 36 (1907), 126-134.
  • [9] A. Pizer, On the $ 2$-part of the class number of imaginary quadratic number fields, J. Number Theory 8 (1976), 184-192. MR 0406975 (53:10759)
  • [10] Kenneth S. Williams, The class number of $ Q(\sqrt p )$ modulo 4, for $ p \equiv 5(\mod 8)$ a prime, Pacific J. Math. 92 (1981), 241-248. MR 618061 (82i:12008)
  • [11] -, The class number of $ Q(\sqrt { - 2p} )$ modulo 8, for $ p \equiv 5(\mod 8)$ a prime, Rocky Mountain J. Math. 11 (1981), 19-26. MR 612125 (83b:12009)
  • [12] Kenneth S. Williams and James D. Currie, Class numbers and biquadratic reciprocity, Canad. J. Math. 34 (1982), 969-988. MR 672691 (84b:12014)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1985-0770517-0
Keywords: Quadratic fields, class number, fundamental unit, Dirichlet class number formula
Article copyright: © Copyright 1985 American Mathematical Society

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