120 PRINCIPAL THEOREM ON ALGEBRAS [chap, viii 
76. Division algebras as direct sums of simple matric 
algebras. 
Theorem. If D is a division algebra over a non- 
modular field F, there exist a finite number of roots of equa 
tions with coefficients in F whose adjunction to F gives a 
field F 1 such that the algebra D 1 over F 1} which has the same 
basal units as D, is a direct sum of simple matric algebras 
over F t . 
Select any element x of D not the product of the 
modulus e by a number of F. By § 60, x is a root of 
either characteristic equation, and hence of a certain 
equation ^>(co)=o of minimum degree s> 1 having 
coefficients in F. 
Let F' be the field obtained by adjoining to F all the 
roots X x , . . . . , X, of 4>{co) =0. Let D' be the algebra 
over F' having the same basal units as D. Then 
(x—\e) .... (x—\ s e) = <t>(x)= o 
in D'. Since x is not the product of e by a number X; 
of F' (footnote in § 74), no one of the x — \e is zero, and 
yet their product is zero. Hence D f is not a division 
algebra by Theorem 4 of § 43. 
The division algebra D is simple (§52). Hence by 
§ 74 D' is semi-simple and (§ 40) is either simple or a 
direct sum of simple algebras over F'. Each such simple 
algebra is the direct product of a division algebra A by 
a simple matric algebra, each over F' (§51). The order 
of each Di is less than that of D'; this is evident for the 
second case in which D' was a direct sum, and also for 
the first case in which D' was simple, provided the matric 
factor is of order >1; but the remaining case i — 1, 
D'=D t , is excluded since D' is not a division algebra.