Given a diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: To devise a process according to which it can be determined by a finite number of operations whether the equation is solvable in rational integers. 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. 12. EXTENSION OF KRONECKER'S THEOREM ON ABELIAN FIELDS TO ANY ALGEBRAIC REALM OF RATIONALITY. The theorem that every abelian number field arises from the realm of rational numbers by the composition of fields of roots of unity is due to Kronecker. This fundamental theorem in the theory of integral equations contains two statements, namely: First. It answers the question as to the number and existence of those [Pg 23] equations which have a given degree, a given abelian group and a given discriminant with respect to the realm of rational numbers. Second. It states that the roots of such equations form a realm of algebraic numbers which coincides with the realm obtained by assigning to the argument in the exponential function all rational numerical values in succession. The first statement is concerned with the question of the determination of certain algebraic numbers by their groups and their branching. This question corresponds, therefore, to the known problem of the determination of algebraic functions corresponding to given Riemann surfaces. The second statement furnishes the required numbers by transcendental means, namely, by the exponential function .