The Euclidean condition in pure cubic and complex quartic fields

Cioffari, Vincent G.

doi:10.1090/S0025-5718-1979-0514835-5

The Euclidean condition in pure cubic and complex quartic fields
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by Vincent G. Cioffari PDF

Math. Comp. 33 (1979), 389-398 Request permission

Abstract:

In this paper we prove that a field $Q{(^3}\sqrt d )$ is euclidean with respect to the ordinary norm if and only if $d = 2,3$ or 10. We also prove that certain fields of the form $Q{(^4}\sqrt { - d} ),d > 0$, are or are not euclidean.

References

J. W. S. Cassels, The inhomogeneous minimum of binary quadratic, ternary cubic and quaternary quartic forms, Proc. Cambridge Philos. Soc. 48 (1952), 72–86. MR 47709, DOI 10.1017/s0305004100027389
H. J. Godwin, On Euclid’s algorithm in some cubic fields with signature one, Quart. J. Math. Oxford Ser. (2) 18 (1967), 333–338. MR 219509, DOI 10.1093/qmath/18.1.333
Elizabeth M. Taylor, Euclid’s algorithm in cubic fields with complex conjugates, J. London Math. Soc. (2) 14 (1976), no. 1, 49–54. MR 419399, DOI 10.1112/jlms/s2-14.1.49

C. R. Acad. Sci. Paris

Richard B. Lakein, Euclid’s algorithm in complex quartic fields, Acta Arith. 20 (1972), 393–400. MR 304350, DOI 10.4064/aa-20-4-393-400