402 KANSAS UNIVERSITY SCIENCE BULLETIN. 



83. An integral algebraic number satisfies an equa- 

 tion 



x''-ha,x''~'+ . . . +a,a;^-^ + . . . 4-a, = o, 

 where every a^ is an integer. A good illustration of 

 the serviceableness of the group theory process for 

 arithmetic is the theorem : The whole numbers of a 

 quadratic number body form an abelian group on addi- 

 tion and, if we exclude zero, an abelian semigroup with 

 regard to multiplication. 



84. A more striking illustration is furnished by 

 Dedekinds' ''Ideals." An ideal is defined as a combi- 

 nation of the whole numbers of a number body posses- 

 sing the first group property for both addition and 

 multiplication. The ideals of a given body form an 

 abelian semigroup with respect to multiplication. 

 There is no addition of ideals. 



85. Similar applications are found in the expression 

 of an ideal by aid of its basis and in the cognate formu- 

 lation of the transformations of a collineation in space 

 of N dimensions. The latter question resolves itself 

 into that of complex numbers with N principal units. 

 Here it is no longer possible to preserve even the re- 

 stricted abelian semigroup on the higher rule. 



Not only is the abelian character lost, as in quater- 

 nions, but the semigroup property may be violated on 

 account of the possibility of a combination by the 

 higher rule being zero when none of the factors is zero. 



86. It is now possible to describe more concisely the 

 relation of the transfinite arithmetic to common arith- 

 metic. The symbols defined in § 22 form a set of num- 

 bers with infinitely many principal units w, vf, vf, . . 

 i. e., the principal units form a series of type w. In 

 finite arithmetic, it is true, the symbols u^, u^, . . . all 

 belong to the set generated from u as principal unit, 

 but if we allow this analogy in the transfinite system 

 there will be still more new symbols. 



