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Cubic forms over algebraic number fields

Published online by Cambridge University Press:  26 February 2010

D. J. Lewis
D. J. Lewis
Affiliation:
University of Notre Dame, Notre Dame, Indiana, U.S.A.
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Extract

It is our purpose here to show that, using results already in the literature, it is easy to prove the following and similar theorems.

For every positive integer d, there exists an integer Ψ (d) such that if K is an algebraic number field of degree d over the field of rational numbers then every cubic form f(x1 x2, …, xn) over K, with n ≥ Ψ(d), has a non-trivial zero in K.

Type
Research Article
Information
Mathematika , Volume 4 , Issue 2 , December 1957 , pp. 97 - 101
Copyright
Copyright © University College London 1957

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References

1.Brauer, Richard, “A note on systems of homogeneous algebraic equations”, Bull. American Math. Soc., 51 (1945), 749–755.CrossRefGoogle Scholar
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3.Lewis, D. J., “Cubic congruences”, Michigan J. of Math., 4 (1957), 85—95.CrossRefGoogle Scholar
4.Peck, L. G., “Diophantine equations in algebraic number fields”, American J. of Math., 71 (1949), 387–402.CrossRefGoogle Scholar