At the center of this chapter stands the *Wedderburn structure theorem*, according to which every *simple artinian algebra* is isomorphic to a matrix algebra Mn(D) over some division algebra D, with n and (the isomorphism class of) D uniquely determined. A structure result in abstract algebra, and a very satisfying one at that, which one can prove through simple methods of *linear algebra*! (This was first done by E. Artin.) It reduces the study of simple artinian algebras to that of *division algebras* and thus represents not only an achievement but also a starting point for further investigations, in that it leads us to pursue a classification of division algebras. This problem turns out to be tougher than it may appear at first, even after making further restrictions; nonetheless we will be able to deal in Chapter 31 with the case of *local* division algebras.

## Keywords

Tensor Product Characteristic Polynomial Division Algebra Matrix Algebra Simple Algebra## Preview

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