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).
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 $<n$ 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.
