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Mathematical Problems

Chapter 12: 11. QUADRATIC FORMS WITH ANY ALGEBRAIC NUMERICAL COEFFICIENTS.
By David Hilbert
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About This Book

A systematic collection of fundamental open problems is presented to chart directions for future mathematical research. Topics range across set theory, the consistency of arithmetic, geometry, transformation groups, number theory, algebraic forms, analysis, the calculus of variations, and the mathematical formulation of physical axioms. Challenges highlighted include questions about the continuum, irrationality and transcendence, the distribution and properties of primes, reciprocity laws, solvability of Diophantine equations, uniformization and boundary-value problems, and the finiteness or structure of function systems. Each problem is posed to emphasize its technical difficulty and its capacity to stimulate new methods and broader theoretical development.

11. QUADRATIC FORMS WITH ANY ALGEBRAIC NUMERICAL COEFFICIENTS.

Our present knowledge of the theory of quadratic number fields[25] puts us in a position to attack successfully the theory of quadratic forms with any number of variables and with any algebraic numerical coefficients. This leads in particular to the interesting problem: to solve a given quadratic equation with algebraic numerical coefficients in any number of variables by integral or fractional numbers belonging to the algebraic realm of rationality determined by the coefficients.

The following important problem may form a transition to algebra and the theory of functions:

[25] Hilbert, "Ueber den Dirichlet'schen biquadratischen Zahlenkörper," Math. Annalen, vol. 45; "Ueber die Theorie der relativquadratischen Zahlenkörper," Jahresber. d. Deutschen Mathematiker-Vereinigung, 1897, and Math. Annalen, vol. 51; "Ueber die Theorie der relativ-Abelschen Körper," Nachrichten d. K. Ges. d. Wiss. zu Göttingen, 1898; Grundlagen der Geometrie, Leipzig, 1899, Chap. VIII, § 83 [Translation by Townsend, Chicago, 1902]. Cf. also the dissertation of G. Rückle, Göttingen, 1901.