The solvability of the equation $ax^{2}+by^{2}=c$ in quadratic fields
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- by Neal Plotkin PDF
- Proc. Amer. Math. Soc. 34 (1972), 337-339 Request permission
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
- Burton Fein, Basil Gordon, and John H. Smith, On the representation of $-1$ as a sum of two squares in an algebraic number field, J. Number Theory 3 (1971), 310–315. MR 319940, DOI 10.1016/0022-314X(71)90005-9
- Burton W. Jones, The Arithmetic Theory of Quadratic Forms, Carcus Monograph Series, no. 10, Mathematical Association of America, Buffalo, N.Y., 1950. MR 0037321, DOI 10.5948/UPO9781614440109
- L. J. Mordell, Diophantine equations, Pure and Applied Mathematics, Vol. 30, Academic Press, London-New York, 1969. MR 0249355
Additional Information
- © Copyright 1972 American Mathematical Society
- 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