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The inhomogeneous minima of complex cubic norm forms

Published online by Cambridge University Press:  24 October 2008

H. P. F. Swinnerton-Dyer
H. P. F. Swinnerton-Dyer
Affiliation:
Trinity College Cambridge
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Extract

1. Let K1, K2, K3 be three conjugate* cubic algebraic number fields, of which we shall take K1 to be real and K2, K3 to be complex conjugate. Let ωi1, ωi2, ωi3 the conjugate basis for the integers of Ki. Write

so that ξi runs through the integers of Ki as the xj run independently through the rational integers, and the ξi (i = 1, 2, 3) are conjugate. The determinant of the ξi, regarded as linear forms in the xj, is ± √d, where d is the common discriminant of the fields.

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Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 50 , Issue 2 , April 1954 , pp. 209 - 219
Copyright
Copyright © Cambridge Philosophical Society 1954

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References

REFERENCES

(1) Barnes, E. S. and Swinnerton-Dyer, H. P. F. The inhomogeneous minima of binary quadratic forms. I. Acta Math., Stockh., 87 (1952), 259–323.Google Scholar
(2) Clarke, L. E. Non-homogeneous linear forms associated with algebraic fields. Quart. J. Math. (2), 2 (1951), 308–15.Google Scholar
(3) Davenport, H. On the product of n linear forms. Proc. Camb. phil. Soc. 49 (1953), 190–3.Google Scholar