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Mathematics of Computation

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Class numbers of quadratic fields determined by solvability of Diophantine equations

Author: R. A. Mollin
Journal: Math. Comp. 48 (1987), 233-242
MSC: Primary 11R11; Secondary 11R29
MathSciNet review: 866112
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Abstract: In the literature there has been considerable attention given to the exploration of relationships between certain diophantine equations and class numbers of quadratic fields. In this paper we provide criteria for the insolvability of certain diophantine equations. This result is then used to determine when related real quadratic fields have class number bigger than 1. Moreover, based on criteria which we find for the solvability of a certain class of diophantine equations, we are able to determine when the class number of related imaginary quadratic fields is divisible by a given integer.

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Keywords: Quadratic field, class number, diophantine equation, unit
Article copyright: © Copyright 1987 American Mathematical Society