{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T03:11:24Z","timestamp":1776827484220,"version":"3.51.2"},"reference-count":13,"publisher":"American Mathematical Society (AMS)","issue":"239","license":[{"start":{"date-parts":[[2003,1,11]],"date-time":"2003-01-11T00:00:00Z","timestamp":1042243200000},"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                    In this paper, we are interested in solving the so-called norm equation\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"script upper N Subscript upper L slash upper K Baseline left-parenthesis x right-parenthesis equals a\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                  <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">N<\/mml:mi>\n                                <\/mml:mrow>\n                              <\/mml:mrow>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>L<\/mml:mi>\n                                <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                  <mml:mo>\/<\/mml:mo>\n                                <\/mml:mrow>\n                                <mml:mi>K<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mi>a<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">{\\mathcal N}_{L\/K} (x)=a<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , where\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L slash upper K\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>L<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\/<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mi>K<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">L\/K<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is a given arbitrary extension of number fields and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"a\">\n                        <mml:semantics>\n                          <mml:mi>a<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">a<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    a given algebraic number of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper K\">\n                        <mml:semantics>\n                          <mml:mi>K<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">K<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . By considering\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper S\">\n                        <mml:semantics>\n                          <mml:mi>S<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">S<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -units and relative class groups, we show that if there exists at least one solution (in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L\">\n                        <mml:semantics>\n                          <mml:mi>L<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">L<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , but not necessarily in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"double-struck upper Z Subscript upper L\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi mathvariant=\"double-struck\">Z<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:mrow>\n                            <mml:mi>L<\/mml:mi>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">{\\mathbb Z}_L<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ), then there exists a solution for which we can describe precisely its prime ideal factorization. In fact, we prove that under some explicit conditions, the\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper S\">\n                        <mml:semantics>\n                          <mml:mi>S<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">S<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -units that are norms are norms of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper S\">\n                        <mml:semantics>\n                          <mml:mi>S<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">S<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -units. This allows us to limit the search for rational solutions to a finite number of tests, and we give the corresponding algorithm. When\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"a\">\n                        <mml:semantics>\n                          <mml:mi>a<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">a<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is an algebraic integer, we also study the existence of an integral solution, and we can adapt the algorithm to this case.\n                  <\/p>","DOI":"10.1090\/s0025-5718-02-01309-1","type":"journal-article","created":{"date-parts":[[2002,9,20]],"date-time":"2002-09-20T15:46:54Z","timestamp":1032536814000},"page":"1287-1305","source":"Crossref","is-referenced-by-count":18,"title":["Solving norm equations in relative number fields using \ud835\udc46-units"],"prefix":"10.1090","volume":"71","author":[{"given":"Denis","family":"Simon","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2002,1,11]]},"reference":[{"issue":"3","key":"1","doi-asserted-by":"publisher","first-page":"191","DOI":"10.1007\/BF01428941","article-title":"\u00dcber Normen algebraischer Zahlen","volume":"251","author":"Bartels, Hans-Jochen","year":"1980","journal-title":"Math. Ann.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5831","issn-type":"print"},{"key":"2","series-title":"Graduate Texts in Mathematics","isbn-type":"print","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4684-9327-6","volume-title":"Cohomology of groups","volume":"87","author":"Brown, Kenneth S.","year":"1982","ISBN":"https:\/\/id.crossref.org\/isbn\/0387906886"},{"key":"3","unstructured":"[3] C. Chevalley: Sur la th\u00e9orie du corps de classe dans les corps finis et les corps locaux, J. Fac. Sci Tokyo, 2 (1933) 365-475."},{"key":"4","series-title":"Graduate Texts in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-02945-9","volume-title":"A course in computational algebraic number theory","volume":"138","author":"Cohen, Henri","year":"1993","ISBN":"https:\/\/id.crossref.org\/isbn\/3540556400"},{"key":"5","isbn-type":"print","doi-asserted-by":"publisher","first-page":"433","DOI":"10.1007\/BFb0054882","article-title":"Computation of relative quadratic class groups","author":"Cohen, Henri","year":"1998","ISBN":"https:\/\/id.crossref.org\/isbn\/3540646574"},{"key":"6","unstructured":"[6] H. Cohen, F. Diaz y Diaz, M. Olivier: Algorithms for finite abelian groups, submitted to J. Symb. Comp."},{"key":"7","unstructured":"[7] C. Fieker : Ueber Relative Normgleichungen in Algebraischen Zahlk\u00f6rpern, Dissertation, Technische Univertit\u00e4t Berlin (1997);"},{"issue":"217","key":"8","doi-asserted-by":"publisher","first-page":"399","DOI":"10.1090\/S0025-5718-97-00761-8","article-title":"On solving relative norm equations in algebraic number fields","volume":"66","author":"Fieker, C.","year":"1997","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"9","isbn-type":"print","doi-asserted-by":"publisher","first-page":"194","DOI":"10.1007\/3-540-12868-9_103","article-title":"A procedure for determining algebraic integers of given norm","author":"Fincke, U.","year":"1983","ISBN":"https:\/\/id.crossref.org\/isbn\/3540128689"},{"key":"10","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1515\/crll.1980.316.1","article-title":"An algorithm for finding an algebraic number whose norm is a given rational number","volume":"316","author":"Garbanati, Dennis A.","year":"1980","journal-title":"J. Reine Angew. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0075-4102","issn-type":"print"},{"key":"11","series-title":"Publications de l'Universit\\'{e} de Nancago, No. VIII","volume-title":"Corps locaux","author":"Serre, Jean-Pierre","year":"1968"},{"key":"12","first-page":"197","article-title":"Normen algebraischer Zahlen","author":"Siegel, C. L.","year":"1973","journal-title":"Nachr. Akad. Wiss. G\\\"{o}ttingen Math.-Phys. Kl. II","ISSN":"https:\/\/id.crossref.org\/issn\/0065-5295","issn-type":"print"},{"key":"13","unstructured":"[13] D. Simon: \u00c9quations dans les Corps de Nombres et Discriminants Minimaux, th\u00e8se, Universit\u00e9 de Bordeaux I (1998)."}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2002-71-239\/S0025-5718-02-01309-1\/S0025-5718-02-01309-1.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2002-71-239\/S0025-5718-02-01309-1\/S0025-5718-02-01309-1.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T23:00:53Z","timestamp":1776726053000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2002-71-239\/S0025-5718-02-01309-1\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2002,1,11]]},"references-count":13,"journal-issue":{"issue":"239","published-print":{"date-parts":[[2002,7]]}},"alternative-id":["S0025-5718-02-01309-1"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-02-01309-1","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":[[2002,1,11]]}}}