The definition of a real division algebra is actually well-established, though there are a number of very closely related concepts (like real composition algebra, real associative algebra, real alternative algebra) that are sometimes not distinguished carefully. Four names are associated with results closely related to the 1, 2, 4, 8 theorem (1958) that the only real division algebras have those dimensions. But as far as I can tell the result is actually due (independently) to John Milnor and Michel Kervaire. The two others are Raoul Bott, whose periodicity theorem (1957, 1959) was used in these proofs, and J. Frank Adams, who proved a closely related theorem, the non-existence of elements of Hopf invariant one (1960). (It's sometimes stated incorrectly that the only real division algebras are the reals, complexes, quaternions, and octonions. But in fact without more restrictions there are other real division algebras in some of these dimensions. Not in dimension 1, where any real division algebra is isomorphic to the reals, but others do exist in dimension 2; I'm not sure about 4 and 8.) The very best source for all these concepts is imho the really terrific book "Numbers" by Ebbinghaus et al., 1991. The following is taken straight from that book, p. 183ff. -------------- Real Algebras. A vector space V over R with a "product mapping" (or multiplication) V x V —> V with (x,y) |—> xy is said to be an algebra over R (or an R-algebra, or a real algebra), if the two distributive laws (ax + by)z = a(xz) + b(yz), x(ay + bz) = a(xy) + b(xz) hold for all a, b in R and x, y, z in V (bilinearity of the product). . . . . . . An algebra A (that is not the trivial 0 algebra) is said to be a "division algebra" if for all a, b in V, with a != 0, the two equations ax = b and ya = b have unique solutions in A. --------------------------- Interestingly, so far the only proofs of the 1, 2, 4, 8 theorem use topology rather heavily, and afaik no purely algebraic proof has been found. —Dan
On Dec 27, 2015, at 10:15 AM, Warren D Smith <warren.wds@gmail.com> wrote:
Dan said the only real division algebras are in dimensions 1,2,4,8. However, there actually is one in dimension 16 too, depending on the precise technical definition of "division algebra."
That is explained in paper #73 here: http://rangevoting.org/WarrenSmithPages/homepage/works.html
It would be rather nice if anybody figured out a use for it.