# Elementary classification of division rings

Are there examples (other than the two mentioned below) of fields $$K$$ such that the classification of all finite dimensional division $$K$$-algebras is possible using only elementary theory (lets say a basic course in algebra, including field and galois theory)?

The only examples I am aware of are finite fields and $$\mathbb{R}$$ (and trivially algebraically closed fields).

• You ask if examples exist, and then you list some, so of course you are aware they exist. Do you mean to ask whether others exist? – Wojowu Apr 6 at 13:36
• Probably $\mathbf{C}(\!(t)\!)$ is not hard either. – YCor Apr 6 at 14:04

The classification is trivial for a $$\,C_1$$-field $$K$$ (that is, such that any homogeneous polynomial in $$K[x_1,\ldots ,x_n]$$ of degree $$ has a nontrivial zero): the only such division algebras are the finite extensions of $$K$$. For the proof you just need the notion of reduced norm, which can be explained in a reasonably elementary way (see Central simple algebra).
$$C_1$$-fields include finite fields and extensions of transcendance degree 1 of an algebraically closed field (Tsen's theorem); again, the proof in each case is relatively elementary.