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On Euclid's Algorithm in cubic self-conjugate fields

Published online by Cambridge University Press:  24 October 2008

H. Heilbronn
H. Heilbronn
Affiliation:
The Royal FortBristol 8
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Extract

In a paper published in these Proceedings I proved that there are only a finite number of quadratic fields in which Euclid's Algorithm (E.A.) holds. Recently Davenport has found a new proof of this theorem based on the theory of the minima of the product of linear inhomogeneous forms.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 46 , Issue 3 , July 1950 , pp. 377 - 382
Copyright
Copyright © Cambridge Philosophical Society 1950

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References

† Vol. 34 (1938), 521–6.

‡ Proc. London Math. Soc. (in course of publication).Google Scholar

§ A preliminary account of the cubic case is given in C.R. Acad. Sci., Paris, 228 (1949), 883–5.Google Scholar Detailed proofs will appear in Acta Mathematica. I am indebted to Prof. Davenport for a private communication of his results.

† See Lemma 4 of paper referred to in the firet footnote to p. 377.

† We preserve the convention that small italics denote positive rational integers, but d need no longer be a prime, p continues to denote rational primes.

‡ We define α1 ≡ α2 (mod m), if two integers β1, β2 exist such that β1 ≡ β2 ≡ 1 (mod m) in the usual sense and α1β1 = α2β2. We call a number prime to d, if it is the quotient of two integers which are both prime to d.