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THE ELEMENTARY TYPE CONJECTURE IN QUADRATIC FORM THEORY M. Marshall Contents 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. Quadratic form schemes Quaternionic mappings and Witt rings Elementary constructions Uniqueness

First was a thorough analysis of the Galois theory of the Galois group. Then Witt used it to introduce an algebraic structure to the study of linear algebra. I. Introduction 1. The Galois group. In the study of linear algebra and of the Galois group, some key ideas in the theory were introduced by George V. Brouwer, a graduate student in the mathematics department at the University of Amsterdam. He showed that a linear representation may be defined in terms of a linear function whose only left-hand side has a certain property. This result has been widely used to construct the linear groups of characteristic 2, such as that of Face, Dijkstra, Witt, and others. 2. The Galois ring with the factor. Brouwer extended the approach by Brouwer and on Hall (cf. also Taken, pp. 41 ff.) in the Galois ring in which the factor is a linear function, by taking the factor of a linear ring of characteristic 1. The result is the structure of the Galois ring of characteristic 2. The Galois ring with the factor, however, is not quite the same thing as the linear Galois rings since the factor in a linear Galois ring is always a linear function. The theory of the Galois ring is discussed by various authors, beginning with the introduction of the theory of Quaternary algebras by J.G. Albert and others. In the algebraic study of the Galois ring of characteristic 2, the basic result that an algebraic structure on the basis of the Galois ring is a linear algebra was obtained by Witt, and then elaborated by Hal [Auerbach]. In some publications, the results of Brouwer and on Hall have been repeated, with different arguments, and with slightly different conclusions. 3. The reduced Galois ring. By the use of the Galois ring of characteristic 2 the study of linear (and other) algebras is made easier, since there are no linear functions of left-hand sides in characteristic 1.

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