The reals as an algebra over the rationals

Question

R, the real numbers, is an infinite dimensional commutative division algebra over the rationals Q. Is there an example of an infinite dimensional noncommutative division algebra over the rationals Q?

Answer 1

Sure: take any noncommutative division algebra over $\mathbb{R}$, like the quaternions. They inherit a $\mathbb{Q}$-algebra structure from the embedding $\mathbb{Q}\rightarrow \mathbb{R}$, and they're clearly infinite dimensional, since $\mathbb{R}$ is already infinite dimensional.