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

Mollin, R. A.

doi:10.1090/S0002-9939-1988-0915707-1

Necessary and sufficient conditions for the class number of a real quadratic field to be one, and a conjecture of S. Chowla
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Proc. Amer. Math. Soc. 102 (1988), 17-21 Request permission

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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6

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Eindeutigkeit der Zerlegung in Primzahlfaktoren in Quadratischen Zahlkörpern

142

Sur la résolution des équations

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