78 I The Expanding Universe of Numbers
Of wider significance are the associative algebras introduced in 1878 by
Clifford [15] (pp. 266–276) as a common generalization of quaternions and Grassmann
algebra.Clifford algebraswere used by Lipschitz (1886) to represent orthogonal trans-
formations inn-dimensional space. There is an extensive discussion of Clifford alge-
bras in Deheuvels [20]. For their applications in physics, see Salingaros and Wene [64].
Proposition 32 has many uses. The proof given here is extracted from Nagahara
and Tominaga [55].
It was proved by both Kervaire (1958) and Milnor (1958) that if a division
algebraA(not necessarily associative) contains the real fieldRin its centre and is of
finite dimension as a vector space overR, then this dimension must be 1,2,4 or 8
(but the algebra need not be isomorphic toR,C,HorO). All known proofs use deep
results from algebraic topology, which was first applied to the problem by H. Hopf
(1940). For more information about the proof, see Chapter 11 (by Hirzebruch) of
Ebbinghauset al.[24].
When is the product of two sums of squares again a sum of squares? To make
the question precise, call a triple (r,s,t) of positive integers ‘admissible’ if there
exist real numbersρijk( 1 ≤i≤t, 1 ≤ j≤r, 1 ≤k ≤s)such that, for every
x=(ξ 1 ,...,ξr)∈Rrand everyy=(η 1 ,...,ηs)∈Rs,
(ξ 12 +···+ξr^2 )(η^21 +···+η^2 s)=ζ 12 +···+ζt^2 ,where
ζi=∑rj= 1∑sk= 1ρijkξjηk.The question then becomes, which triples(r,s,t)are admissible? It is obvious
that( 1 , 1 , 1 )is admissible and the relationn(x)n(y)=n(xy)for the norms of com-
plex numbers, quaternions and octonions shows that(t,t,t)is admissible also for
t= 2 , 4 ,8. It was proved by Hurwitz (1898) that(t,t,t)is admissible for no other
values oft. A survey of the general problem is given by Shapiro [65].
General introductions to algebra are provided by Birkhoff and MacLane [7] and
Herstein [35]. More extended treatments are given in Jacobson [41] and Lang [48].
The theory of groups is treated in M. Hall [29] and Rotman [60]. An especially
significant class of groups is studied in Humphreys [38].
IfHis a subgroup of a finite groupG, then it is possible to choose a system of left
coset representatives ofHwhich is also a system of right coset representatives. This
interesting, but not very useful, fact belongs to combinatorics rather than to group the-
ory. We mention it because it was the motivation for the theorem of P. Hall (1935) on
systems of distinct representatives, also known as the ‘marriage theorem’. Further de-
velopments are described in Mirsky [53]. For quantitative versions, with applications
to operations research, see Ford and Fulkerson [27].
The theory of rings separates into two parts. Noncommutative ring theory, which
now incorporates the structure theory of associative algebras, is studied in the books
of Herstein [36], Kasch [42] and Lam [46]. Commutative ring theory, which grew
out of algebraic number theory and algebraic geometry, is studied in Atiyah and
Macdonald [4] and Kunz [45].