A computable formula for the class number of the imaginary quadratic field $ \mathbb Q(\sqrt{-p}), \ p = 4n-1 $

  • Received: 01 May 2021 Revised: 01 July 2021 Published: 07 September 2021
  • Primary: 11E41, 11R29; Secondary: 11E16

  • Using elementary methods, we count the quadratic residues of a prime number of the form $ p = 4n-1 $ in a manner that has not been explored before. The simplicity of the pattern found leads to a novel formula for the class number $ h $ of the imaginary quadratic field $ \mathbb Q(\sqrt{-p}). $ Such formula is computable and does not rely on the Dirichlet character or the Kronecker symbol at all. Examples are provided and formulas for the sum of the quadratic residues are also found.

    Citation: Jorge Garcia Villeda. A computable formula for the class number of the imaginary quadratic field $ \mathbb Q(\sqrt{-p}), \ p = 4n-1 $[J]. Electronic Research Archive, 2021, 29(6): 3853-3865. doi: 10.3934/era.2021065

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  • Using elementary methods, we count the quadratic residues of a prime number of the form $ p = 4n-1 $ in a manner that has not been explored before. The simplicity of the pattern found leads to a novel formula for the class number $ h $ of the imaginary quadratic field $ \mathbb Q(\sqrt{-p}). $ Such formula is computable and does not rely on the Dirichlet character or the Kronecker symbol at all. Examples are provided and formulas for the sum of the quadratic residues are also found.



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    [1] H. Cohen, A Course in Computational Algebraic Number Theory, Volume 138 of Graduate Text in Mathematics, Springer-Verlag, New York, 1993. doi: 10.1007/978-3-662-02945-9
    [2] L. E. Dickson, Introduction to the Theory of Numbers, Dover Publ. Inc., New York, 1957.
    [3] P. G. L. Dirichlet, Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält, Volume 1, Cambridge University Press, (2012), 313–342. doi: 10.1017/CBO9781139237338.023
    [4] J. Garcia, Sum of quadratic-type residues modulus a prime $p = 4n-1$, work in progress.
    [5] W. Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers, Springer, 2004. doi: 10.1007/978-3-662-07001-7
    [6] D. Shanks, Class number, a theory of factorization, and genera, 1969 Number Theory Institute (Proc. Sympos. Pure Math., Volume XX, State Univ. New York, Stony Brook, N.Y., 1969), Amer. Math. Soc., (1971), 415–440.
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