{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T03:35:37Z","timestamp":1776828937765,"version":"3.51.2"},"reference-count":35,"publisher":"American Mathematical Society (AMS)","issue":"242","license":[{"start":{"date-parts":[[2003,10,17]],"date-time":"2003-10-17T00:00:00Z","timestamp":1066348800000},"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                    We prove that there is precisely one normal CM-field of degree 48 with class number one which has a normal CM-subfield of degree 16: the narrow Hilbert class field of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"double-struck upper Q left-parenthesis StartRoot 5 EndRoot comma StartRoot 101 EndRoot comma theta right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"double-struck\">Q<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msqrt>\n                              <mml:mn>5<\/mml:mn>\n                            <\/mml:msqrt>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:msqrt>\n                              <mml:mn>101<\/mml:mn>\n                            <\/mml:msqrt>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>\n                              \u03b8\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathbb {Q}(\\sqrt {5}, \\sqrt {101}, \\theta )<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    with\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"theta cubed minus theta squared minus 5 theta minus 1 equals 0\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msup>\n                              <mml:mi>\n                                \u03b8\n                                \n                              <\/mml:mi>\n                              <mml:mn>3<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>\n                                \u03b8\n                                \n                              <\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mn>5<\/mml:mn>\n                            <mml:mi>\n                              \u03b8\n                              \n                            <\/mml:mi>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\theta ^3 - \\theta ^2 -5 \\theta -1=0<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    .\n                  <\/p>","DOI":"10.1090\/s0025-5718-02-01443-6","type":"journal-article","created":{"date-parts":[[2003,2,10]],"date-time":"2003-02-10T11:10:52Z","timestamp":1044875452000},"page":"1003-1017","source":"Crossref","is-referenced-by-count":3,"title":["The class number one problem for some non-abelian normal CM-fields of degree 48"],"prefix":"10.1090","volume":"72","author":[{"given":"Ku-Young","family":"Chang","sequence":"first","affiliation":[]},{"given":"Soun-Hi","family":"Kwon","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2002,10,17]]},"reference":[{"key":"1","series-title":"Series in Pure Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1142\/0663","volume-title":"Class number parity","volume":"8","author":"Conner, P. E.","year":"1988","ISBN":"https:\/\/id.crossref.org\/isbn\/9971506696"},{"issue":"9","key":"2","doi-asserted-by":"publisher","first-page":"2517","DOI":"10.1090\/S0002-9939-00-05555-6","article-title":"Class numbers of imaginary abelian number fields","volume":"128","author":"Chang, Ku-Young","year":"2000","journal-title":"Proc. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9939","issn-type":"print"},{"key":"3","unstructured":"[CK2] K.-Y. Chang and S.-H. Kwon, CM-fields of degree 2\ud835\udc5d\ud835\udc5e, preprint."},{"key":"4","doi-asserted-by":"crossref","unstructured":"[CK3] K.-Y. Chang and S.-H. Kwon, The non-abelian normal CM-fields of degree 36 with class number one, Acta Arith., 101 (2002), 53-61.","DOI":"10.4064\/aa101-1-4"},{"issue":"1","key":"5","doi-asserted-by":"publisher","first-page":"37","DOI":"10.1007\/BF02139701","article-title":"Some analytic bounds for zeta functions and class numbers","volume":"55","author":"Hoffstein, Jeffrey","year":"1979","journal-title":"Invent. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0020-9910","issn-type":"print"},{"issue":"3","key":"6","doi-asserted-by":"publisher","first-page":"505","DOI":"10.1007\/BF02571442","article-title":"On a ratio between relative class numbers","volume":"211","author":"Horie, Kuniaki","year":"1992","journal-title":"Math. Z.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5874","issn-type":"print"},{"key":"7","series-title":"Cambridge Mathematical Textbooks","isbn-type":"print","volume-title":"Representations and characters of groups","author":"James, Gordon","year":"1993","ISBN":"https:\/\/id.crossref.org\/isbn\/0521440246"},{"issue":"3-4","key":"8","doi-asserted-by":"publisher","first-page":"267","DOI":"10.1006\/jsco.1996.0126","article-title":"KANT V4","volume":"24","author":"Daberkow, M.","year":"1997","journal-title":"J. Symbolic Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0747-7171","issn-type":"print"},{"issue":"1","key":"9","doi-asserted-by":"publisher","first-page":"67","DOI":"10.5802\/aif.1747","article-title":"Corps di\u00e9draux \u00e0 multiplication complexe principaux","volume":"50","author":"Lefeuvre, Yann","year":"2000","journal-title":"Ann. Inst. Fourier (Grenoble)","ISSN":"https:\/\/id.crossref.org\/issn\/0373-0956","issn-type":"print"},{"issue":"2","key":"10","doi-asserted-by":"publisher","first-page":"387","DOI":"10.5802\/jtnb.257","article-title":"The class number one problem for some non-abelian normal CM-fields of degree 24","volume":"11","author":"Lemmermeyer, F.","year":"1999","journal-title":"J. Th\\'{e}or. Nombres Bordeaux","ISSN":"https:\/\/id.crossref.org\/issn\/1246-7405","issn-type":"print"},{"issue":"1","key":"11","doi-asserted-by":"publisher","first-page":"47","DOI":"10.4064\/aa-67-1-47-62","article-title":"Determination of all non-normal quartic CM-fields and of all non-abelian normal octic CM-fields with class number one","volume":"67","author":"Louboutin, St\u00e9phane","year":"1994","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"issue":"3","key":"12","doi-asserted-by":"publisher","first-page":"523","DOI":"10.1112\/S0024611598000318","article-title":"The class number one problem for some non-abelian normal CM-fields of 2-power degrees","volume":"76","author":"Louboutin, St\u00e9phane","year":"1998","journal-title":"Proc. London Math. Soc. (3)","ISSN":"https:\/\/id.crossref.org\/issn\/0024-6115","issn-type":"print"},{"issue":"9","key":"13","doi-asserted-by":"publisher","first-page":"3657","DOI":"10.1090\/S0002-9947-97-01768-6","article-title":"The class number one problem for some non-abelian normal CM-fields","volume":"349","author":"Louboutin, St\u00e9phane","year":"1997","journal-title":"Trans. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9947","issn-type":"print"},{"issue":"2","key":"14","doi-asserted-by":"publisher","first-page":"227","DOI":"10.1112\/jlms\/54.2.227","article-title":"Determination of all quaternion octic CM-fields with class number 2","volume":"54","author":"Louboutin, St\u00e9phane","year":"1996","journal-title":"J. London Math. Soc. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0024-6107","issn-type":"print"},{"issue":"4","key":"15","doi-asserted-by":"publisher","first-page":"634","DOI":"10.1007\/BF02305438","article-title":"On a binary problem","volume":"60","author":"\u00c8minyan, K. M.","year":"1996","journal-title":"Mat. Zametki","ISSN":"https:\/\/id.crossref.org\/issn\/0025-567X","issn-type":"print"},{"issue":"2","key":"16","doi-asserted-by":"publisher","first-page":"173","DOI":"10.4064\/aa-82-2-173-196","article-title":"The class number one problem for the non-abelian normal CM-fields of degree 16","volume":"82","author":"Louboutin, St\u00e9phane","year":"1997","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"issue":"2","key":"17","doi-asserted-by":"publisher","first-page":"259","DOI":"10.4064\/cm-80-2-259-265","article-title":"The class number one problem for the dihedral and dicyclic CM-fields","volume":"80","author":"Louboutin, St\u00e9phane","year":"1999","journal-title":"Colloq. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0010-1354","issn-type":"print"},{"issue":"229","key":"18","doi-asserted-by":"publisher","first-page":"371","DOI":"10.1090\/S0025-5718-99-01096-0","article-title":"Computation of relative class numbers of CM-fields by using Hecke \ud835\udc3f-functions","volume":"69","author":"Louboutin, St\u00e9phane","year":"2000","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"19","isbn-type":"print","doi-asserted-by":"publisher","first-page":"413","DOI":"10.1007\/10722028_26","article-title":"Fast computation of relative class numbers of CM-fields","author":"Louboutin, St\u00e9phane","year":"2000","ISBN":"https:\/\/id.crossref.org\/isbn\/3540676953"},{"issue":"3-4","key":"20","doi-asserted-by":"publisher","first-page":"283","DOI":"10.5486\/pmd.2000.2165","article-title":"Class number problems for dicyclic CM-fields","volume":"57","author":"Louboutin, S.","year":"2000","journal-title":"Publ. Math. Debrecen","ISSN":"https:\/\/id.crossref.org\/issn\/0033-3883","issn-type":"print"},{"key":"21","unstructured":"[LP2] S. Louboutin and Y.-H. Park, The Class number one problem for the non-abelian normal CM-fields of degree 42, preprint."},{"issue":"3","key":"22","doi-asserted-by":"publisher","first-page":"201","DOI":"10.4064\/aa-89-3-201-215","article-title":"Construction of the real dihedral number fields of degree 2\ud835\udc5d. Applications","volume":"89","author":"Louboutin, St\u00e9phane","year":"1999","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"issue":"3","key":"23","first-page":"297","article-title":"Class numbers of real cyclic number fields with small conductor","volume":"37","author":"Masley, John Myron","year":"1978","journal-title":"Compositio Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0010-437X","issn-type":"print"},{"key":"24","series-title":"Lecture Notes in Mathematics","isbn-type":"print","doi-asserted-by":"crossref","DOI":"10.1007\/BFb0088938","volume-title":"The determination of units in real cyclic sextic fields","volume":"797","author":"M\u00e4ki, Sirpa","year":"1980","ISBN":"https:\/\/id.crossref.org\/isbn\/3540099840"},{"key":"25","doi-asserted-by":"crossref","first-page":"1","DOI":"10.5802\/aif.307","article-title":"Sur l\u2019arithm\u00e9tique des extensions galoisiennes \u00e0 groupe de Galois di\u00e9dral d\u2019ordre 2\ud835\udc5d","volume":"19","author":"Martinet, Jacques","year":"1969","journal-title":"Ann. Inst. Fourier (Grenoble)","ISSN":"https:\/\/id.crossref.org\/issn\/0373-0956","issn-type":"print"},{"issue":"3","key":"26","doi-asserted-by":"publisher","first-page":"275","DOI":"10.1007\/BF01389854","article-title":"Some analytic estimates of class numbers and discriminants","volume":"29","author":"Odlyzko, A. M.","year":"1975","journal-title":"Invent. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0020-9910","issn-type":"print"},{"issue":"4","key":"27","doi-asserted-by":"publisher","first-page":"319","DOI":"10.4064\/aa-92-4-319-338","article-title":"Inclusion of CM-fields and divisibility of relative class numbers","volume":"92","author":"Okazaki, Ryotaro","year":"2000","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"key":"28","doi-asserted-by":"crossref","unstructured":"[P] Y.-H. Park, The class number one problem for the non-abelian normal CM-fields of degree 24 and 40, Acta Arith., 101 (2002), 63-80.","DOI":"10.4064\/aa101-1-5"},{"key":"29","unstructured":"[Pa] C. Batut, K. Belabas, D. Bernardi, H. Cohen, M. Olivier, Pari-GP version 2.0.11."},{"issue":"3","key":"30","doi-asserted-by":"publisher","first-page":"211","DOI":"10.4064\/aa-83-3-211-223","article-title":"Determination of all non-quadratic imaginary cyclic number fields of 2-power degree with relative class number \u226420","volume":"83","author":"Park, Young-Ho","year":"1998","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"key":"31","unstructured":"[PsK] S.-M. Park and S.-H. Kwon, The class number one problem for normal CM-fields of degree 32 II, preprint."},{"key":"32","series-title":"Graduate Texts in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-1934-7","volume-title":"Introduction to cyclotomic fields","volume":"83","author":"Washington, Lawrence C.","year":"1997","ISBN":"https:\/\/id.crossref.org\/isbn\/0387947620","edition":"2"},{"issue":"206","key":"33","doi-asserted-by":"publisher","first-page":"899","DOI":"10.2307\/2153549","article-title":"The determination of the imaginary abelian number fields with class number one","volume":"62","author":"Yamamura, Ken","year":"1994","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"2","key":"34","doi-asserted-by":"publisher","first-page":"175","DOI":"10.1006\/jnth.1999.2421","article-title":"The non-normal quartic CM-fields and the octic dihedral CM-fields with relative class number two","volume":"79","author":"Yang, Hee-Sun","year":"1999","journal-title":"J. Number Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0022-314X","issn-type":"print"},{"key":"35","unstructured":"[YPK] H.-S. Yang, S.-M. Park and S.-H. Kwon, Class number one problem for normal CM-fields of degree 32, preprint."}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2003-72-242\/S0025-5718-02-01443-6\/S0025-5718-02-01443-6.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2003-72-242\/S0025-5718-02-01443-6\/S0025-5718-02-01443-6.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T23:15:23Z","timestamp":1776726923000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2003-72-242\/S0025-5718-02-01443-6\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2002,10,17]]},"references-count":35,"journal-issue":{"issue":"242","published-print":{"date-parts":[[2003,4]]}},"alternative-id":["S0025-5718-02-01443-6"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-02-01443-6","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":[[2002,10,17]]}}}