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Abstract Quadratic Forms

Published online by Cambridge University Press:  20 November 2018

Irving Kaplansky  and
Richard J. Shaker
Irving Kaplansky
Affiliation:
University of Chicago, Chicago, Illinois
Richard J. Shaker
Affiliation:
University of Chicago, Chicago, Illinois
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We shall be studying the following structure, which we shall call a V-form (“Vector-valued form”). Let G and W be additive abelian groups with every element of order 2 (i.e. vector spaces over the field GF(2) of two elements). Let there be given a symmetric bilinear map from G × G to W; we shall write it simply as a product ab. We define an equivalence relation on unordered n-ples of G. For n = 2: (a, b) ~ (c, d) if a + b = c + d and ab = cd. For n > 2 we define equivalence “piecewise”: there is to be a chain from (a1, …, an) to (b1, …, bn) where at each step only two elements are changed in accordance with the equivalence just defined for n = 2.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 1218 - 1233
Copyright
Copyright © Canadian Mathematical Society 1969

References

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