{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T20:13:38Z","timestamp":1776802418453,"version":"3.51.2"},"reference-count":19,"publisher":"American Mathematical Society (AMS)","issue":"307","license":[{"start":{"date-parts":[[2018,2,16]],"date-time":"2018-02-16T00:00:00Z","timestamp":1518739200000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    Conditionally on the Generalized Riemann Hypothesis (GRH), we prove the following results: (1) a cyclic number field of degree\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"5\">\n                        <mml:semantics>\n                          <mml:mn>5<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">5<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is norm-Euclidean if and only if\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"normal upper Delta equals 11 Superscript 4 Baseline comma 31 Superscript 4 Baseline comma 41 Superscript 4\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"normal\">\n                              \u0394\n                              \n                            <\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:msup>\n                              <mml:mn>11<\/mml:mn>\n                              <mml:mn>4<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:msup>\n                              <mml:mn>31<\/mml:mn>\n                              <mml:mn>4<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:msup>\n                              <mml:mn>41<\/mml:mn>\n                              <mml:mn>4<\/mml:mn>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\Delta =11^4,31^4,41^4<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ; (2) a cyclic number field of degree\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"7\">\n                        <mml:semantics>\n                          <mml:mn>7<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">7<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is norm-Euclidean if and only if\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"normal upper Delta equals 29 Superscript 6 Baseline comma 43 Superscript 6\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"normal\">\n                              \u0394\n                              \n                            <\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:msup>\n                              <mml:mn>29<\/mml:mn>\n                              <mml:mn>6<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:msup>\n                              <mml:mn>43<\/mml:mn>\n                              <mml:mn>6<\/mml:mn>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\Delta =29^6,43^6<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ; (3) there are no norm-Euclidean cyclic number fields of degrees\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"19\">\n                        <mml:semantics>\n                          <mml:mn>19<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">19<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ,\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"31\">\n                        <mml:semantics>\n                          <mml:mn>31<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">31<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ,\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"37\">\n                        <mml:semantics>\n                          <mml:mn>37<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">37<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ,\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"43\">\n                        <mml:semantics>\n                          <mml:mn>43<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">43<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ,\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"47\">\n                        <mml:semantics>\n                          <mml:mn>47<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">47<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ,\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"59\">\n                        <mml:semantics>\n                          <mml:mn>59<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">59<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ,\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"67\">\n                        <mml:semantics>\n                          <mml:mn>67<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">67<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ,\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"71\">\n                        <mml:semantics>\n                          <mml:mn>71<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">71<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ,\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"73\">\n                        <mml:semantics>\n                          <mml:mn>73<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">73<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ,\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"79\">\n                        <mml:semantics>\n                          <mml:mn>79<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">79<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ,\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"97\">\n                        <mml:semantics>\n                          <mml:mn>97<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">97<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    .\n                  <\/p>\n                  <p>\n                    Our proofs contain a large computational component, including the calculation of the Euclidean minimum in some cases; the correctness of these calculations does not depend upon the GRH. Finally, we improve on what is known unconditionally in the cubic case by showing that any norm-Euclidean cyclic cubic field must have conductor\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"f less-than-or-equal-to 157\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>f<\/mml:mi>\n                            <mml:mo>\n                              \u2264\n                              \n                            <\/mml:mo>\n                            <mml:mn>157<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">f\\leq 157<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    except possibly when\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"f element-of left-parenthesis 2 dot 10 Superscript 14 Baseline comma 10 Superscript 50 Baseline right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>f<\/mml:mi>\n                            <mml:mo>\n                              \u2208\n                              \n                            <\/mml:mo>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mo>\n                              \u22c5\n                              \n                            <\/mml:mo>\n                            <mml:msup>\n                              <mml:mn>10<\/mml:mn>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>14<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:msup>\n                              <mml:mn>10<\/mml:mn>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>50<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">f\\in (2\\cdot 10^{14}, 10^{50})<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    .\n                  <\/p>","DOI":"10.1090\/mcom\/3169","type":"journal-article","created":{"date-parts":[[2016,5,4]],"date-time":"2016-05-04T10:00:27Z","timestamp":1462356027000},"page":"2535-2549","source":"Crossref","is-referenced-by-count":4,"title":["The Euclidean algorithm in quintic and septic cyclic fields"],"prefix":"10.1090","volume":"86","author":[{"given":"Pierre","family":"Lezowski","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Kevin","family":"McGown","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2017,2,16]]},"reference":[{"issue":"191","key":"1","doi-asserted-by":"publisher","first-page":"355","DOI":"10.2307\/2008811","article-title":"Explicit bounds for primality testing and related problems","volume":"55","author":"Bach, Eric","year":"1990","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"2","doi-asserted-by":"publisher","first-page":"259","DOI":"10.1007\/BF02392288","article-title":"The inhomogeneous minima of binary quadratic forms. I","volume":"87","author":"Barnes, E. S.","year":"1952","journal-title":"Acta Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0001-5962","issn-type":"print"},{"issue":"259","key":"3","doi-asserted-by":"publisher","first-page":"1547","DOI":"10.1090\/S0025-5718-07-01932-1","article-title":"Euclidean minima of totally real number fields: algorithmic determination","volume":"76","author":"Cerri, Jean-Paul","year":"2007","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"4","doi-asserted-by":"publisher","first-page":"289","DOI":"10.4153\/cjm-1950-026-7","article-title":"Euclid\u2019s algorithm in real quadratic fields","volume":"2","author":"Chatland, H.","year":"1950","journal-title":"Canad. J. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0008-414X","issn-type":"print"},{"key":"5","doi-asserted-by":"publisher","first-page":"333","DOI":"10.1093\/qmath\/18.1.333","article-title":"On Euclid\u2019s algorithm in some cubic fields with signature one","volume":"18","author":"Godwin, H. J.","year":"1967","journal-title":"Quart. J. Math. Oxford Ser. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0033-5606","issn-type":"print"},{"key":"6","doi-asserted-by":"publisher","first-page":"699","DOI":"10.1112\/jlms\/s1-40.1.699","article-title":"On Euclid\u2019s algorithm in some quartic and quintic fields","volume":"40","author":"Godwin, H. J.","year":"1965","journal-title":"J. London Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0024-6107","issn-type":"print"},{"issue":"201","key":"7","doi-asserted-by":"publisher","first-page":"421","DOI":"10.2307\/2153178","article-title":"On the Euclidean nature of four cyclic cubic fields","volume":"60","author":"Godwin, H. J.","year":"1993","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"8","doi-asserted-by":"publisher","first-page":"257","DOI":"10.4153\/cjm-1951-029-4","article-title":"On Euclid\u2019s algorithm in cyclic fields","volume":"3","author":"Heilbronn, H.","year":"1951","journal-title":"Canad. J. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0008-414X","issn-type":"print"},{"key":"9","doi-asserted-by":"publisher","first-page":"377","DOI":"10.1017\/s0305004100025883","article-title":"On Euclid\u2019s algorithm in cubic self-conjugate fields","volume":"46","author":"Heilbronn, H.","year":"1950","journal-title":"Proc. Cambridge Philos. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0008-1981","issn-type":"print"},{"issue":"5","key":"10","first-page":"385","article-title":"The Euclidean algorithm in algebraic number fields","volume":"13","author":"Lemmermeyer, Franz","year":"1995","journal-title":"Exposition. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0723-0869","issn-type":"print"},{"issue":"287","key":"11","doi-asserted-by":"publisher","first-page":"1397","DOI":"10.1090\/S0025-5718-2013-02746-9","article-title":"Computation of the Euclidean minimum of algebraic number fields","volume":"83","author":"Lezowski, Pierre","year":"2014","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"1","key":"12","doi-asserted-by":"publisher","first-page":"227","DOI":"10.1142\/S1793042112500133","article-title":"Norm-Euclidean cyclic fields of prime degree","volume":"8","author":"McGown, Kevin J.","year":"2012","journal-title":"Int. J. Number Theory","ISSN":"https:\/\/id.crossref.org\/issn\/1793-0421","issn-type":"print"},{"issue":"2","key":"13","doi-asserted-by":"publisher","first-page":"425","DOI":"10.5802\/jtnb.804","article-title":"Norm-Euclidean Galois fields and the generalized Riemann hypothesis","volume":"24","author":"McGown, Kevin J.","year":"2012","journal-title":"J. Th\\'{e}or. Nombres Bordeaux","ISSN":"https:\/\/id.crossref.org\/issn\/1246-7405","issn-type":"print"},{"issue":"4","key":"14","doi-asserted-by":"publisher","first-page":"1289","DOI":"10.1016\/j.jnt.2012.09.011","article-title":"On the second smallest prime non-residue","volume":"133","author":"McGown, Kevin J.","year":"2013","journal-title":"J. Number Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0022-314X","issn-type":"print"},{"key":"15","doi-asserted-by":"publisher","first-page":"577","DOI":"10.1112\/jlms\/s1-44.1.577","article-title":"On Euclid\u2019s algorithm in some cyclic cubic fields","volume":"44","author":"Smith, J. R.","year":"1969","journal-title":"J. London Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0024-6107","issn-type":"print"},{"key":"16","doi-asserted-by":"publisher","first-page":"317","DOI":"10.2307\/2004927","article-title":"Calculation of the gamma function by Stirling\u2019s formula","volume":"25","author":"Spira, Robert","year":"1971","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"5","key":"17","doi-asserted-by":"publisher","first-page":"1653","DOI":"10.1142\/S1793042115400163","article-title":"The Burgess inequality and the least \ud835\udc58th power non-residue","volume":"11","author":"Trevi\u00f1o, Enrique","year":"2015","journal-title":"Int. J. Number Theory","ISSN":"https:\/\/id.crossref.org\/issn\/1793-0421","issn-type":"print"},{"issue":"1","key":"18","first-page":"56","article-title":"On the maximum number of consecutive integers on which a character is constant","volume":"2","author":"Trevi\u00f1o, Enrique","year":"2012","journal-title":"Mosc. J. Comb. Number Theory","ISSN":"https:\/\/id.crossref.org\/issn\/2220-5438","issn-type":"print"},{"key":"19","doi-asserted-by":"publisher","first-page":"201","DOI":"10.1016\/j.jnt.2014.10.019","article-title":"The least \ud835\udc58-th power non-residue","volume":"149","author":"Trevi\u00f1o, Enrique","year":"2015","journal-title":"J. Number Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0022-314X","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2017-86-307\/S0025-5718-2017-03169-0\/S0025-5718-2017-03169-0.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2017-86-307\/S0025-5718-2017-03169-0\/S0025-5718-2017-03169-0.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T19:18:45Z","timestamp":1776799125000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2017-86-307\/S0025-5718-2017-03169-0\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,2,16]]},"references-count":19,"journal-issue":{"issue":"307","published-print":{"date-parts":[[2017,9]]}},"alternative-id":["S0025-5718-2017-03169-0"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/3169","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":[[2017,2,16]]}}}