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Introduction to Abstract Algebra, 4th Edition by W. Keith Nicholson

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Chapter 11

Finiteness Conditions for Rings and Modules

A scientist worthy of the name, above all a mathematician, experiences in his work the same impression as an artist; his pleasure is as great and of the same nature.

—Henri Poincaré

The field img of complex numbers is a two-dimensional vector space over img A ring that is also a vector space over a field F is called an algebra over F. In the nineteenth century many attempts were made to describe the division algebras that are finite dimensional over img and in particular those (like img) that are fields. Certainly the three-dimensional examples would have had applications to physics, but after looking in vain for such an algebra, the first success came in 1843 when W.R. Hamilton discovered the ring img of quaternions, a four-dimensional algebra that, surprisingly, was not commutative. It was not until 1878 that G. Frobenius showed that there is no three-dimensional example, and that the only possible associative examples are , and Meanwhile, G. ...

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