JOURNAL OP Till. COLLEGE OF SCIENCE, TOKYO IMPERIAL UNIVERSITY. 



VOL. XXXVI, ARTICLE 1. 



to Professor Rikitaro Fujisawa 



os the occasion commemorating his twenty-five years' service 



dedicated by 



his devoted pupil, the author. 



On the Classes of Congruent Integers in an 

 Algebraic Körper. 



By 



Tanzo Takenouchi, Bigalcushi, 



Professor in the Fifth (Kumamoto) HiijlL School. 



Introduction 



Let m be an ideal in an algebraic Icörper. All the integers in 

 the harper can be classified into classes of congruent integers with 

 respect to the modulus nt. By integer, unless specified, we 

 mean a general algebraic integer. If a and a' be any two integers 

 of class A, and ß and ß' be those of class B, the products aß and 



aß' always belong to one and the same class say C. In 



this sense these classes can be composed by multiplication, and we 

 write AB = C. When A and B are given, G is uniquely deter- 

 mined, and hereby the commutative and the associative laws 

 evidently hold. If we consider only those classes which consist 

 of integers relatively prime to m. then, in addition to the above, 



