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

Jorge Garcia Villeda; Jorge Garcia Villeda

doi:10.3934/era.2021065

Electronic Research Archive

2021, Volume 29, Issue 6: 3853-3865. doi: 10.3934/era.2021065

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

Jorge Garcia Villeda

One University Drive, Camarillo, CA 93012, USA

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.
- Class number,
- quadratic residues,
- imaginary fields,
- quadratic field,
- sum of quadratic residues
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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Abstract

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.

References

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[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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