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The class number of $ Q(\surd -p)$, for $ P\equiv 1 ({\rm mod}\ 8)$ a prime


Author: Ezra Brown
Journal: Proc. Amer. Math. Soc. 31 (1972), 381-383
MSC: Primary 10.66
DOI: https://doi.org/10.1090/S0002-9939-1972-0289455-6
MathSciNet review: 0289455
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Abstract: Let $ h( - p)$ be the class number of the quadratic field $ Q(\surd - p)$, where $ p \equiv 1\pmod 8$ is a prime. Write $ p = {a^2} + {b^2} = 2{e^2} - {d^2}$, where $ a \equiv e \equiv d \equiv b + 1 \equiv 1\pmod 2$ and $ e > 0$. We prove that $ h( - p) \equiv 0$ or $ 4\pmod 8$ according as $ (e\vert p) = 1$ or $ - 1$; using this, we prove that $ h( - p) \equiv (p - 1)/2 + b\pmod 8$. The proofs are elementary, relying on the theory of composition of binary quadratic forms.


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DOI: https://doi.org/10.1090/S0002-9939-1972-0289455-6
Keywords: Class number, quadratic residues, quadratic fields, algebraic number fields, composition of binary quadratic forms
Article copyright: © Copyright 1972 American Mathematical Society