Division Algebras of Finite Rank

Abstract

Wedderburn’s theorem entirely reduces the study of semisimple algebras of finite rank over a field K to that of division algebras of finite rank over the same field. We now concentrate on this problem. If D is a division algebra of finite rank over K and L the centre of D then L is a finite extension of K and we can consider D as an algebra over L. Hence the problem divides into two: to study finite field extensions, which is a question of commutative algebra or Galois theory, and that of division algebras of finite rank over a field which is its centre. If an algebra D of finite rank over a field K has K as its centre, then we say that D is a central algebra over K.