Quadratic Forms over Fields
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The algebraic theory of quadratic forms originated in a classical paper by E. Witt . The importance of this paper consists of three essential contributions to the theory. First of all Witt introduced into the theory the geometrical language which is now commonly adopted. Secondly he constructed, in a canonical fashion, a commutative ring from the collection of all regular symmetric bilinear forms over a given field. This construction proved to be of fundamental importance because it allowed mathematicians to ask new and very fruitful questions: What is the structure of this ring and what does it tell us about the forms over the given field? Finally Witt summarized, unified, and extended the then known classification theorems. Despite the landmark importance of Witt’s paper, it was only after an incubation period of almost 30 years that a vigorous development of the algebraic theory of quadratic forms began. This new development started with the appearance of Pfister’s work in 1965 and 1966 which contains above all the first deep structure theorems about the Witt ring. The beauty and the elegance of these results led immediately to new questions, problems, and results and the theory has been flourishing ever since.
KeywordsQuadratic Form Prime Ideal Ring Homomorphism Quaternion Algebra Finite Extension
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