PROCEEDINGS 
OF THE 
NATIONAL ACADEMY OF SCIENCES 
Volume 7 APRIL 15, 1921 Number 4 
QUATERNIONS AND THEIR GENERALIZATIONS 
By Leonard Eugene Dickson 
Department of Mathematics, University of Chicago 
Read before the Academy, April 26, 1921 
1. The discovery of quaternions by W. R. Hamilton in 1843 has led 
to an extensive theory of linear algebras (or closed systems of hyper- 
complex numbers) in which the quaternion algebra plays an important 
r61e. Frobenius 1 proved that the only real linear associative algebras 
in which a product is zero only when one factor is zero are the real number 
system, the ordinary complex number system, and the algebra of real 
quaternions. A much simpler proof has been given by the writer. 2 Later, 
the writer 3 was led to quaternions very naturally by means of the four- 
parameter continuous group which leaves unaltered each line of a set 
of rulings on the quadric surface x\ + x\ -f- %\ + x\ = 0. 
The object of the present note is to derive the algebra of quaternions 
and its direct generalizations without assuming the associative or com- 
mutative law. I shall obtain this interesting result by two distinct 
methods. 
2. The term field will be employed here to designate any set of ordinary 
complex numbers which is closed under addition, subtraction, multipli- 
cation, and division. Thus all complex numbers form a field, likewise 
all real numbers, or all rational numbers. 
Just as a couple (a, b) of real numbers defines an ordinary complex 
number a + bi, where i 2 = —1, so also an n-tuple {x\, . ., x n ) of numbers 
of a field F defines a hypercomplex number 
% = xid + x 2 e 2 + + x n e n , (1) 
where the units e\, . . . , e n are linearly independent with respect to the 
field F and possess a multiplaction table 
n 
e i e J * 2 7 ***** J ' = lf ' ' M )' ( 2 ) 
k = 1 
109 
