NORMALIZED UNITS 
175 
§ 100] 
99. The fundamental theorem on arithmetics of 
algebras. The proof (§ 104) for any rational algebra 
depends upon that for the complex algebra with the same 
basal units. Hence we shall first deduce from the 
general theory of algebras a set of normalized basal 
units of any complex algebra and derive its characteristic 
determinants by a method far simpler than that employed 
by Cartan.* Moreover, our notations are more explicit 
and hence more satisfactory. 
It is only incidental to the goal of rational algebras 
that we find the integral elements of a normalized 
complex algebra. That result alone would not dispose 
of the question for all rational algebras since not all 
types of the latter are rational sub-algebras of complex 
algebras in canonical forms obtained by applying trans 
formations of units with complex coefficients. 
100. Normalized basal units of a nilpotent algebra. 
Lemma. Any associative algebra A of index a is a 
sum of a linear sets B 1 , . . . . , B a , no two with an 
element in common, such that 
(14) BpB q SBp+qA-Bp+q+xA- .... +J5 a (/>+g<a), 
(15) BpBq^Ba (p+q^a). 
For, we may select in turn linear sets B lt B 2 , . . . . 
such that 
A = B I +A 2 , A 2 =B 2 +Af .... , A*~ z =B a - t +A*, A a =B a , 
where B{/\A i+1 = 0 in A { =BiA-A i+I . Thusib^Hb For 
i<j^a, 
Bj^A j ^A i+I , Bi^Bj=o. 
* Annates Fac. Sc. Toulouse, Vol. XII (1898). See the author’s 
Linear Algebras (1914), pp. 44-55.