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Are the 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).

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).

Are the examples 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).

Are the 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).