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 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).