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2023•arXiv (Cornell University)Open access

Class Number of the Imaginary Quadratic Field and Quadratic Residues Identities

Jorge García

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Abstract

A formula for the sum of quadratic residues modulus a prime $p=4n-1$ is studied. We relate some terms on this formula with roots of quadratics and provide an exhaustive analysis of new concepts based on these roots. A number of formulas for the sum of the quadratic residues are obtained. We finalize the paper by obtaining several identities involving $h(-p)$ the class number of the imaginary quadratic field $\mathbb Q(\sqrt{-p}).$

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A formula for the sum of quadratic residues modulus a prime $p=4n-1$ is studied. We relate some terms on this formula with roots of quadratics and provide an exhaustive analysis of new concepts based on these roots. A number of formulas for the sum of the quadratic residues are obtained. We finalize the paper by obtaining several identities involving $h(-p)$ the class number of the imaginary quadratic field $\mathbb Q(\sqrt{-p}).$

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Available abstract

A formula for the sum of quadratic residues modulus a prime $p=4n-1$ is studied. We relate some terms on this formula with roots of quadratics and provide an exhaustive analysis of new concepts based on these roots. A number of formulas for the sum of the quadratic residues are obtained. We finalize the paper by obtaining several identities involving $h(-p)$ the class number of the imaginary quadratic field $\mathbb Q(\sqrt{-p}).$

Key concepts: Quadratic field, Quadratic equation, Quadratic residue, Mathematics, Binary quadratic form, The Imaginary, Class (philosophy), Class number

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