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Cubic forms over algebraic number fields

Published online by Cambridge University Press:  24 October 2008

C. P. Ramanujam
C. P. Ramanujam
Affiliation:
Tata Institute of Fundamental Research, Bombay
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Extract

Davenport has proved (3) that any cubic form in 32 or more variables with rational coefficients has a non-trivial rational zero. He has also announced that he has subsequently been able to reduce the number of variables to 29. Following the method of (3), we shall prove that any cubic form over any algebraic number field has a non-trivial zero in that field, provided that the number of variables is at least 54. The following is the precise form of our result.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 59 , Issue 4 , October 1963 , pp. 683 - 705
Copyright
Copyright © Cambridge Philosophical Society 1963

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References

REFERENCES

(1)Birch, B. J.Waring's problem over algebraic number fields. Proc. Cambridge Philos. Soc. 57 (1961), 449–459CrossRefGoogle Scholar
(2)Birch, B. J. and Lewis, D. J.P-adic forms. J. Indian Math. Soc. 23 (1959), 11–33Google Scholar
(3)Davenport, H.Cubic forms in thirty-two variables. Philos. Trans. Roy. Soc. London Ser. A, 251 (1958–1959), 193–232.Google Scholar
(4)Hua, L. K.On exponential sums over an algebraic number field. Canadian J. Math. 3 (1951), 44–51CrossRefGoogle Scholar