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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Necessary and sufficient conditions for the class number of a real quadratic field to be one, and a conjecture of S. Chowla


Author: R. A. Mollin
Journal: Proc. Amer. Math. Soc. 102 (1988), 17-21
MSC: Primary 11R29,; Secondary 11R11
DOI: https://doi.org/10.1090/S0002-9939-1988-0915707-1
MathSciNet review: 915707
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Abstract: Based on the fundamental unit of $ Q\left( {\sqrt n } \right)$, an arbitrary real quadratic field, we provide a necessary condition for the class number $ h\left( n \right)$ to be 1. For $ n = 4{m^2} + 1$ we prove the equivalence of three necessary and sufficient conditions for $ h\left( n \right)$ to be 1. One of these conditions is that $ - {x^2} + x + {m^2}$ is prime for all integers $ x$ such that $ 1 < x < m$. This is the exact analogue of the complex quadratic field case. We discuss the connection with a conjecture of S. Chowla as well as with other related topics.


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DOI: https://doi.org/10.1090/S0002-9939-1988-0915707-1
Keywords: Class number 1, real quadratic field, prime valued polynomial
Article copyright: © Copyright 1988 American Mathematical Society