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Algebraic integral representations by arbitrary forms

Published online by Cambridge University Press:  26 February 2010

E. C. Dade
Affiliation:
California Institute of Technology, Pasadena
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If(X1, …, Xn), n ≥ 3, is a non-singular quadratic form with rational integral coefficients whose greatest common divisor is 1, then G. L. Watson [1] showed that f(x1, …, xn) = 1, for suitable algebraic integers x1, …, xn. In the present paper we extend this result to forms of arbitrary degree, with algebraic integers as coefficients (see Theorem 3). In fact we prove the stronger result (Theorem 2) that, if f(X1, …, Xn) is any polynomial with relatively prime algebraic integers as coefficients, then f(x1, …, xn) is a unit, for suitable algebraic integers x1, …, xn. Unfortunately, our result is just an existence theorem. We cannot limit the size of the field which x1, …, xn generate, as Watson could.

Type
Research Article
Copyright
Copyright © University College London 1963

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References

1.Watson, G. L., “A problem of Dade on quadratic forms”, Mathematika, 10 (1963), 101106.Google Scholar
2.Hecke, E., Vorlesungen über die Theorie der algebraischen Zahlen (Leipzig 1923), p. 121, Satz 98.Google Scholar
3.Steinitz, E., “Rechteckige Systems und Moduln in algebraischon Zahlkörpern I”, Math. Annalen, 17 (1911), p. 340.Google Scholar