The Group of Classes of Congruent Quadratic Integers with Respect to a Composite Ideal Modulus
Arthur Ranum · Transactions of the American Mathematical Society · 1910
If in the ordinary theory of rational numbers we consider a composite integer m as modulus, and if from among the classes of congruent integers with respect to that modulus we select those which are prime to the modulus, they form a well-known multiplicative group, which has been called by Weber (Algebra, vol.2, 2d edition, p. 60), the most important example of a finite abelian group.In the more general theory of numbers in an algebraic field we may in a corresponding manner take as modulus a composite ideal, which includes as a special case a composite principal ideal, that is, an integer in the field, and if we regard all 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 abelian group f under *The reader of this book should be on the lookout for a number of minor errors and some rather misleading statements.Reference will also be made to Hilbert's Bericht in the Jahresbericht der Deutschen Mathematiker-Vereinigung, vol. 4 (1897), pages 181-194, for the source of much of the material in Sommer's book, and to Bachmann's Neuere Zahlentheorie, Sammlung Schubert, for an introduction to the theory of algebraic numbers, in which the author, like Sommer, confines himself to the quadratic field.t Sommer 1. c, pp.78-81.