23/03/13 21:13:18.99 UeELXD7y.net
>>359
つづき
If n > 2, then D cannot be a division algebra. Assume that n > 2. Let u = e1e2en. It is easy to see that u2 = 1 (this only works if n > 2). If D were a division algebra, 0 = u2 ? 1 = (u ? 1)(u + 1) implies u = ±1, which in turn means: en = ?e1e2 and so e1, ..., en?1 generate D. This contradicts the minimality of W.
Remarks and related results
The fact that D is generated by e1, ..., en subject to the above relations means that D is the Clifford algebra of Rn. The last step shows that the only real Clifford algebras which are division algebras are Cｌ0, Cｌ1 and Cｌ2.
As a consequence, the only commutative division algebras are R and C. Also note that H is not a C-algebra. If it were, then the center of H has to contain C, but the center of H is R. Therefore, the only finite-dimensional division algebra over C is C