Sum of 16 squares?

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}?

Solution 1:

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.

Solution 2:

After googling some more I've found that not only does the sum of squares not exist for N=16 or N=2^k, the only possible values are just N=1,2,4,8

This theorem, known as Hurwitz theorem, was a vital step in the classification of normed division algebras.

See: http://en.wikipedia.org/wiki/Normed_division_algebra

EDIT: this is wrong, see above.