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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.


References [Enhancements On Off] (What's this?)

  • [1] B. Fein and B. Gordon, On the representation of $ - 1$ as a sum of two squares in an algebraic number field, J. Number Theory 3 (1971), 310-315. MR 0319940 (47:8481)
  • [2] B. W. Jones, The arithmetic theory of quadratic forms, Carus Math. Monograph Series, no. 10, Math. Assoc. of Amer., distributed by Wiley, New York, 1950. MR 12, 244. MR 0037321 (12:244a)
  • [3] L. J. Mordell, Diophantine equations, Pure and Appl. Math., vol. 30, Academic Press, New York, 1969. MR 40 #2600. MR 0249355 (40:2600)

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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

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