There is a finite field for every power of a prime integer, the field of rational numbers is countable, and R and C are fields with continuum cardinals. No non-commutative division ring can be finite, and the quaternions are a non-commutative division ring with continuum cardinal. So I would like to have examples of some countable non-commutative division rings
0
Add a comment |
1 Answer
$\begingroup$ $\endgroup$
For a concrete example, take the "rational quaternions," i.e. $$\{a+bi+cj+dk: a,b,c,d\in\mathbb{Q}\}.$$ (Or the "algebraic quaternions," or etc.)
More generally, any uncountable division ring $R$ with fixed noncommuting elements $a,b$ has a smallest sub-division ring $R'\ni a,b$, and this $R'$ is always countable; we can explicitly construct $R'$ in stages, as follows:
Let $R_0$ be the subring of $R$ generated by $a$ and $b$.
Having defined $R_i$, let $R_{i+1}$ be the ring generated by $R_i\cup\{{1\over r}: 0\not=r\in R_i\}$.
Our ultimate subring $R'=\bigcup_{i\in\mathbb{N}}R_i$ is then a countable union of countable subrings, and so countable.
Even more generally, a similar strategy applies to any first-order-expressible theory (in a countable language); this is the downward Lowenheim-Skolem theorem.
