Basic Notions of Algebra pp 90-96 | Cite as

# 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*.

## Keywords

Number Field Division Algebra Isosceles Triangle Finite Rank Simple Central Algebra## Preview

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