$\Bbb R,\Bbb C,\Bbb H$ are the only complete normed division rings

Question

Does anyone have a proof or reference for this often-quoted fact? How complicated is the proof?

If $K$ is a complete normed division ring, then $K$ is isomorphic to the real numbers, complex numbers, or the quaternions.

Answer 1

The statement is false; the fields $\Bbb Q_p$ and $\Bbb C_p$ of $p$-adic rationals and complexes are (metrically) complete normed fields, which are of course not isomorphic to $\Bbb R,\Bbb C,\Bbb H$.

The correct statement along these lines is known as the Frobenius theorem:

Theorem (Frobenius 1877): Every finite-dimensional (associative) division algebra over $\Bbb R$ is isomorphic to one of $\Bbb R,\Bbb C,\Bbb H$.

If one drops the requirement of associativity (and adds an inner product), there is a similar statement which reduces the field to these three, plus the set $\Bbb O$ of octonions, proven by Hurwitz in 1898.

In fact, the assumption that the algebras are finite-dimensional is also unnecessary:

Theorem (Urbanik and Wright 1960): Every (not necessarily associative) normed division algebra over $\Bbb R$ is isomorphic to one of $\Bbb R,\Bbb C,\Bbb H,\Bbb O$.