The equation of Abel's theorem in the theory of algebraic functions expresses, as is well known, the necessary and sufficient condition that the points in question on the Riemann surface are the zero points of an algebraic function belonging to the surface. The exact analogue of Abel's theorem, in the theory of the number field of class , is the equation of the law of quadratic reciprocity[29] which declares that the ideal is then and only then a principal ideal of the number field when the quadratic residue of the number with respect to the ideal is positive. It will be seen that in the problem just sketched the three fundamental branches of mathematics, number theory, algebra and function theory, come into closest touch with one another, and I am certain that the [Pg 25] theory of analytical functions of several variables in particular would be notably enriched if one should succeed in finding and discussing those functions which play the part for any algebraic number field corresponding to that of the exponential function in the field of rational numbers and of the elliptic modular functions in the imaginary quadratic number field. Passing to algebra, I shall mention a problem from the theory of equations and one to which the theory of algebraic invariants has led me. [26] Elliptische Funktionen und algebraische Zahlen. Braunschweig, 1891. [27] Jahresber. d. Deutschen Math-Vereinigung, vol. 6, and an article soon to appear in the Math. Annalen [Vol. 55, p. 301]: "Ueber die Entwickelung der algebraischen Zahlen in Potenzreihen." [28] Math. Annalen vol. 50 (1898). [29] Cf. Hilbert, "Ueber die Theorie der relativ-Abelschen Zahlkörper," Gött. Nachrichten, 1898. 13. IMPOSSIBILITY OF THE SOLUTION OF THE GENERAL EQUATION OF THE 7TH DEGREE BY MEANS OF FUNCTIONS OF ONLY TWO ARGUMENTS.