[FOM] Number theory proof mentioned by Frege

Vaughan Pratt pratt at cs.stanford.edu
Thu Apr 17 16:24:45 EDT 2008

John Baldwin wrote:
> the quaternions lose commutativity, and with more losses you get degree 8 
> and I think 16.  Then no more.

What one gives up here is by no means uniquely determined.  You're 
presumably referring to the octonions, which retain the existence of 
multiplicative inverses of nonzero elements in exchange for giving up 
associativity of multiplication, and the sedenions, which give up 
inverses as well.

For those for whom associativity is important and who are willing to 
accept a few zero divisors (as anyone accepting sedenions must), there 
are Clifford algebras, very popular with physicists.  Those over the 
reals are the main ones---Clifford algebras over the complex numbers 
amount to a more simply structured subfamily of the real ones since both 
are representable as square real matrices.

Finite-dimensional Clifford algebras are generated by p generators with 
square 1 and q generators with square -1.  The case (p,q) = (0,1) gives 
the complex numbers while (0,2) gives the quaternions, (2,0) gives the 
2x2 real matrices, and (1,1) is isomorphic to (2,0).

The only commutative Clifford algebras are (0,0) (the reals), (0,1) (the 
complex numbers), and direct sums thereof (so-called by analogy with 
vector spaces, they're really direct products of associative algebras) 
which includes (1,0) as the direct sum of (0,0) with itself, aka the 
hyperbolic numbers.

Vaughan Pratt

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