If in the ordinary theory of rational numbers we consider a composite illteger m as modulus, alld if from among the classes of congruent integers with lespect to that modulus we select those which are prime to the lnodtllus, they form a well-knowll multiplicative group, which has been called by WEBER (Alyebra, vol. 2, 2d edition, p. 60), the most ialportant example of a finite abelian group. In the more general theory of numbers in an algebraic field we nsay in a corresponding manner take as modulus a composite ideal, which illeludes as a special case a composite principal ideal, that is, an integer in the field, and if we regard a11 those integers of the field which are congruent to one another with respect to the modulus as forming a class, and if we select those classes whose integers are prime to the modulus, they also will form a finite abeliall groupt under multiplication. The investigation of the nature of this group is the object of the present paper. I shall confine my attention, however, to a quadratic number-field, and shall determine the structure of the group of classes of congruent quadratic integers with respect to any composite ideal modulus whatever. Several distinct cases arise depending on the nature of the prime ideal factors of the modulus; for every case I shall find a complete system of independent generators of the group. Exactly as in the simpler theory of rational numbers it will appear that the solution of the problem depends essentially on the case in whicls the modulus is a prime-power ideal, that is, a power of a prime ideal. The most important {:ase, however, is probably that in which the modulus is a rational principal ideal or in otller words a rational integer; therefore a separate discussion will be given of this case. Allother interesting case is that in which the group is