110 
MATHEMATICS: L. E. DICKSON 
Proc. N. A. S. 
n which the y's are numbers of F. Let %' = be another hypercom- 
plex number whose coordinates x\ are numbers of F. Then shall 
n n n 
%%' = ^ % i*r e i e v % - x ' = 2 ^ *^lf f ' = */ = 2 
t, J - 1 * = 1 *' m 1 
when / is in F, so that multiplication is distributive. Under these assump- 
tions, the set of all numbers (1) with coordinates in F shall be called a 
linear algebra over F. 
3. We assume that £i is a principal unit (modulus), so that e\X = %e\ = % 
for every number % of the algebra, and write 1 for e\. We assume that 
every number of the algebra satisfies a quadratic equation with coefficients 
in F. If e 2 + 2ae + b = 0, (e + a) 2 = a 2 — 6, so that we may take 
the units to be 1, E 2 , . . . , E n , where E\ = s u , a number of F. For i and 
/ distinct and >1, E { =*= F,- satisfies a quadratic, so that (F t - =*= F,-) 2 = 
5,7 + % =*= (F^F, + EjEi) is a linear function of F t - =*= F,. Thus Z^F^ -f- 
F 7 Fj is a linear function of E t + F, and of Ei~Ej, and hence is a number 
2fy = 2sji of F. 
Let U2, . . . , u„ be arbitrary numbers of F and write [7 = ^Lu k E k , Then 
£/ 2 equals 
n 
Q = 
It is a standard theorem that Q can be reduced to Sqzj,- 2 by a linear trans- 
formation w fe = Sa w v/ with coefficients in F and of determinant =1=0. 
Write 
2 aklEk (1 = 2,..., n). 
ei = 
k,l = 2 
Then 1, e 2 , . . e n are linearly independent and may be taken as the new 
units of our algebra over F. Then 
U = 2 WiE* = 2 %J 
k,l 
Hence 
= etej + e/i = 0 (V, / = 2, . . ., n, i 4= /). (3) 
4. Write * = x x + S^,-. Then (x—Xi) 2 = 2^# t 2 . This gives %%' = 
%'x = o", where 
