Why are the only division algebras over the real numbers the real numbers, the complex numbers, and the quaternions?

Why are the only (associative) division algebras over the real numbers the real numbers, the complex numbers, and the quaternions?

Here a division algebra is an associative algebra where every nonzero number is invertible (like a field, but without assuming commutativity of multiplication).

This is an old result proved by Frobenius, but I can’t remember how the argument goes. Anyone have a quick proof?

Answer

Essentially one first proves that any real division algebra D is a Clifford algebra (i.e. it’s generated by elements of some inner product vector space I subject to relations v^2=\langle v, v\rangle): first one splits D as \mathbb R\oplus D_0 where D_0 is the space of elements with Tr=0 and then one observes that minimal polynomial of a traceless element has the form x^2-a=0 (it’s quadratic because it’s irreducible and the coefficient of x is zero because it is the trace). Now it remains to find out which Clifford algebras are division algebras which is pretty straightforward (well, and it follows from the classification of Clifford algebras).

This proof is written in Wikipedia.

Attribution
Source : Link , Question Author : Noah Snyder , Answer Author : Grigory M

Leave a Comment