{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T21:52:38Z","timestamp":1776721958520,"version":"3.51.2"},"reference-count":9,"publisher":"American Mathematical Society (AMS)","issue":"216","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>Weinberger in 1973 has shown that under the Generalized Riemann Hypothesis for Dedekind zeta functions, an algebraic number field with infinite unit group is Euclidean if and only if it is a principal ideal domain. Using a method recently introduced by us, we give two examples of cubic fields which are Euclidean but not norm\u2013Euclidean.<\/p>","DOI":"10.1090\/s0025-5718-96-00764-8","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:44Z","timestamp":1027707284000},"page":"1675-1679","source":"Crossref","is-referenced-by-count":2,"title":["Non-Galois cubic fields which are Euclidean but not norm-Euclidean"],"prefix":"10.1090","volume":"65","author":[{"given":"David","family":"Clark","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[1996]]},"reference":[{"key":"1","first-page":"285","article-title":"Sur les inverses des \u00e9l\u00e9ments d\u00e9rivables dans un anneau abstrait","volume":"209","author":"Hebroni, P.","year":"1939","journal-title":"C. R. Acad. Sci. Paris","ISSN":"https:\/\/id.crossref.org\/issn\/0001-4036","issn-type":"print"},{"key":"2","unstructured":"[2] D.A. Clark, The Euclidean algorithm for Galois extensions of the rational numbers, McGill University, Montr\u00e9al, 1992."},{"issue":"3-4","key":"3","doi-asserted-by":"publisher","first-page":"327","DOI":"10.1007\/BF02567617","article-title":"A quadratic field which is Euclidean but not norm-Euclidean","volume":"83","author":"Clark, David A.","year":"1994","journal-title":"Manuscripta Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-2611","issn-type":"print"},{"key":"4","doi-asserted-by":"crossref","unstructured":"[4] D.A. Clark and M.R. Murty, The Euclidean algorithm in Galois extensions of \u211a, J. Reine Angew. Math. 459 (1995), 151\u2013162.","DOI":"10.1515\/crll.1995.459.151"},{"key":"5","unstructured":"[5] F. Lemmermeyer, The Euclidean algorithm in algebraic number fields, Exposition. Math. 13 (1995), 385\u2013416."},{"key":"6","unstructured":"[6] H.W. Lenstra, Lectures on Euclidean rings, Bielefeld, 1974."},{"key":"7","unstructured":"[7] J.R. Smith, The inhomogeneous minima of some totally real cubic fields, Computers in Number Theory (A.O.L. Atkin and B.J. Birch, eds.), Academic Press, New York, 1971, pp. 223\u2013224."},{"issue":"1","key":"8","doi-asserted-by":"publisher","first-page":"49","DOI":"10.1112\/jlms\/s2-14.1.49","article-title":"Euclid\u2019s algorithm in cubic fields with complex conjugates","volume":"14","author":"Taylor, Elizabeth M.","year":"1976","journal-title":"J. London Math. Soc. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0024-6107","issn-type":"print"},{"key":"9","first-page":"321","article-title":"On Euclidean rings of algebraic integers","author":"Weinberger, Peter J.","year":"1973"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/1996-65-216\/S0025-5718-96-00764-8\/S0025-5718-96-00764-8.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/1996-65-216\/S0025-5718-96-00764-8\/S0025-5718-96-00764-8.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T21:17:37Z","timestamp":1776719857000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/1996-65-216\/S0025-5718-96-00764-8\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1996]]},"references-count":9,"journal-issue":{"issue":"216","published-print":{"date-parts":[[1996,10]]}},"alternative-id":["S0025-5718-96-00764-8"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-96-00764-8","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[1996]]}}}