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On Euclid's Algorithm in Cyclic Fields

Published online by Cambridge University Press:  20 November 2018

H. Heilbronn
H. Heilbronn*
Affiliation:
The Royal Fort, Bristol8
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In two papers I have proved that there are only a finite number of quadratic fields [6] and of cyclic cubic fields [7] in which Euclid's algorithm (E.A.) holds. Davenport has shown by a different method that there are only a finite number of quadratic fields [1, 2], of non-totally real cubic fields [3, 4] and of totally complex quartic fields in which E.A. holds.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 3 , 1951 , pp. 257 - 268
Copyright
Copyright © Canadian Mathematical Society 1951

References

[1] Davenport, H., Indefinite binary quadratic forms, and Euclid's Algorithm in real quadratic fields, Proc. London Math. Soc, in course of publication.Google Scholar
[2] Davenport, H., Indefinite binary quadratic forms, Quart. J. Math., Oxford Ser. (2), vol. 1 (1950), 54–62.Google Scholar
[3] Davenport, H., Euclid's Algorithm in cubic fields of negative discriminant, Acta Math., vol. 84 (1950), 159–179.Google Scholar
[4] Davenport, H., Euclid's Algorithm in certain quartic fields, Trans. Amer. Math. Soc, vol. 68 (1950), 508–532.Google Scholar
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[6] Heilbronn, H., On Euclid's Algorithm in real quadratic fields, Proc. Cambridge Phil. Soc, vol. 34 (1938), 521–526.Google Scholar
[7] Heilbronn, H., On Euclid's Algorithm in self-conjugate cubic fields, Proc Cambridge Phil. Soc, vol. 46 (1950), 377–382.Google Scholar
[8] Hilbert, D., Bericht ilber die Theorie der algebraischen Zahlkörper, Jber. Deutsch. Math. Verein., vol. 4 (1897), 175–546.Google Scholar
[9] Ingham, A. E., The distribution of prime numbers (Cambridge, 1932).Google Scholar
[10] Polya, G., Über die Verteilung der quadratischen Reste und Nichtreste, Nachr. Akad. Wiss. Göttingen, Math. Phys. Kl. 1918, 21–29.Google Scholar