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A problem of Dade on quadratic forms

Published online by Cambridge University Press:  26 February 2010

G. L. Watson
Affiliation:
University College, London
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Extract

The problem dealt with in this paper was suggested by Dr. E. C. Dade and communicated to me, at Stockholm in August 1962, by Dr. Taussky Todd. It may be stated as follows. Consider the equation

where f is a form (i.e., a homogeneous polynomial) with coefficients in some ring R of algebraic integers. An obviously necessary condition for the solubility of (1) with the xi in R is that the coefficients of f should have no common factor, except for units of R. Now let S be a ring which is an algebraic extension of R, and let us try to solve (1) with the xi in S. The condition just mentioned remains necessary; and Dade has proved in [1] that for some S (depending on R and f) it is sufficient. His theorem is valid for forms of any degree but is merely an existence theorem as far as S is concerned. The problem is to do better for the special case in which f is of degree 2 and R the ring of rational integers; and more precisely, to show if possible that S can be taken to be a quadratic extension R(√q) of R, with an integer q which could be estimated in terms of the coefficients of f.

Type
Research Article
Copyright
Copyright © University College London 1963

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References

1.Dade, E. C., Mathematika, 10 (1963), 96100.CrossRefGoogle Scholar
2.Watson, G. L., Integral quadratic forms (Cambridge Tract No. 51) (Cambridge, 1960).Google Scholar