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Proceedings of the American Mathematical Society

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

 

 

The solvability of the equation $ ax\sp{2}+by\sp{2}=c$ in quadratic fields


Author: Neal Plotkin
Journal: Proc. Amer. Math. Soc. 34 (1972), 337-339
MSC: Primary 12A25
DOI: https://doi.org/10.1090/S0002-9939-1972-0297736-5
MathSciNet review: 0297736
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Abstract: In a recent paper, L. J. Mordell gave necessary and sufficient conditions for the equation $ a{x^2} + b{y^2} = c$ to have algebraic integer solutions in the quadratic field $ Q(\surd ( - n))$. In this paper we drop the requirement that the solutions be algebraic integers. In particular, we prove that $ a{x^2} + b{y^2} = c$ has solutions in $ Q(\surd ( - n))$ if and only if the quadratic form $ ab{t^2} - bc{u^2} - ac{v^2} - n{w^2}$ represents 0 over Q.


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DOI: https://doi.org/10.1090/S0002-9939-1972-0297736-5
Keywords: Diophantine equations, quadratic fields, quadratic forms
Article copyright: © Copyright 1972 American Mathematical Society