In research, much attention is given to organizing knowledge so that it is readily accessible to all those who wish to build upon its foundation. A primary goal for such organization is classifying the objects studied. Biologists have classified the distinguishable kinds of organisms inhabiting the earth and chemists have classified the distinct elements found in the universe. In the 19th century topologists classified 2-dimensional manifolds and in recent years there has been a general consensus that the 3-dimensional manifolds have been classified. Most recently algebraists have classified the finite simple groups. To make order out of the diversity in each area, a method of classification is necessary. Precise definitions and designations of objects are essential in communicating information. The classification of the different types of division algebras is far from being realized, but there is much that can be said. In this talk I will describe the problem of classifying finite dimensional division algebras.
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