TITLE: Sum of 16 squares?
QUESTION [10 upvotes]: It is known that for n=1,2,4,8 the sum of squares is multiplicative. See for instance "Naive Lie theory" Stillwell for a good reference. It happens to be that these identities are the result of the existence of Hypercomplex number systems: The reals, the complex numbers, the quarternions, the octonions. 
Now with the cayley-dickson process one can produce higher 2^k number systems, the first of which the sedenions is 16-dimensional. Is there a sum of squares identity for N=16, or for genereal 2^k?
That is if a={sum of 16 squares}, and b={sum of 16 squares} is ab={sum of 16 squares}?

REPLY [10 votes]: A sum of squares identity does exist for $N = 16$, and in fact when $N$ is any power of 2. For $N$ equal to a power of 2, the set of nonzero sums of $N$ squares in any field $F$ is a subgroup of $F - \{0\}$. 
See http://www.math.uconn.edu/~kconrad/blurbs/linmultialg/pfister.pdf for more, and in particular why this does not violate the Hurwitz theorem.