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A Note on Division Algorithms in Imaginary quadratic Number fields

Published online by Cambridge University Press:  20 November 2018

D. W. Dubois  and
A. Steger
D. W. Dubois
Affiliation:
University of New Mexico
A. Steger
Affiliation:
University of New Mexico
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An integral domain E is said to be Euclidean if there exists a non-negative, integer-valued function g denned on the non-zero elements of E such that for every non-zero x and y in E,

(1) g(xy) ⩾ g(x);

(2) (division algorithm) if x does not divide y then there exists an element q in E, depending on x and y, with .

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 10 , 1958 , pp. 285 - 286
Copyright
Copyright © Canadian Mathematical Society 1958

References

1. Dirichlet, L. and Dedekind, R., Vorlesungen iiber Zahlentheorie (4 Aufl. Braunschweig, 1894).Google Scholar
2. Hardy, G. H. and Wright, E. M., The Theory of Numbers (Oxford, 1954).Google Scholar
3. Hasse, Helmut Ueber eindeutige Zerlegung in Primelemente oder Primhauptideale in Integraetsbereichen, J. reine angew. Math., 159 (1928), 3–12.Google Scholar
4. Van der Waerden, B. L., Modern Algebra (New York, 1949).Google Scholar