A noncommutative analogue of a cyclotomic field extension in the realm of central simple algebras is a Schur algebra, namely a finite dimensional central simple algebra $A$ over a field $K$ which is generated over $K$ by a finite subgroup of the multiplicative group of $A$. Its projective version, a projective Schur algebra, is the noncommutative analogue of a radical field extension, namely a finite dimensional central simple algebra $A$ over a field $K$ which is generated over $K$ by a subgroup $G$ of the multiplicative group of $A$ such that $G/K*$ is finite. The talk will focus on projective Schur algebras and the subgroup of the Brauer group that they determine.