A type of ring that is has all the axioms of a field except that is not commutative.

There are only three finite dimensional division rings over \(\mathbb{R}\), namely \(\mathbb{R}\), \(\mathbb{C}\) and \(\mathbb{H}\).

The only finite dimensional division rings over finite fields are finite fields.

Over \(\mathbb{Q}\), there are infinitely many finite dimensional division algebras. All of them are cyclic division algebras. For a finite dimensional \(\mathbb{Q}\)-division algebra \(D\), \(D\otimes_{\mathbb{Q}} \mathbb{R}\) is always a real semisimple algebra as given there. This is the subject of [1] and [2].

Some interesting results

Siegel mean value theorem

See Siegel mean value theorem for a division algebra variant of the Siegel mean value theorem.

Gram-Schmidt algorithm

A type of Gram-Schmidt process can be done for the \(D_{\mathbb{R}}\).

Minkowski’s theorem

The Minkowski’s theorem on successive minima can be generalized to the setting of free \(D_\mathbb{R}\)-modules.

Heights in grassmanian over division algebras

See [3] and [4] for some nice generalizations of classical Diophantine geometry generalized to division rings.

References