{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T21:48:34Z","timestamp":1776721714671,"version":"3.51.2"},"reference-count":11,"publisher":"American Mathematical Society (AMS)","issue":"213","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    Let\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"script upper F\">\n                        <mml:semantics>\n                          <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                            <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">F<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathcal F<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    be an algebraic number field and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"script upper E\">\n                        <mml:semantics>\n                          <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                            <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">E<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathcal E<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    a quadratic extension with\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"script upper E equals script upper F left-parenthesis StartRoot mu EndRoot right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">E<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">F<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msqrt>\n                              <mml:mi>\n                                \u03bc\n                                \n                              <\/mml:mi>\n                            <\/mml:msqrt>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathcal E=\\mathcal F(\\sqrt {\\mu })<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . We describe a minimal set of elements for generating the integral elements\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"o Subscript script upper E\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>o<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">E<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:mrow>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">o_{\\mathcal E}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"script upper E\">\n                        <mml:semantics>\n                          <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                            <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">E<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathcal E<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    as an\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"o Subscript script upper F\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>o<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">F<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:mrow>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">o_{\\mathcal F}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    module. A consequence of this theoretical result is an algorithm for constructing such a set. The construction yields a simple procedure for computing an integral basis of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"script upper E\">\n                        <mml:semantics>\n                          <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                            <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">E<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathcal E<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    as well. In the last section, we present examples of relative integral bases which were computed with the new algorithm and also give some running times.\n                  <\/p>","DOI":"10.1090\/s0025-5718-96-00686-2","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:44Z","timestamp":1027707284000},"page":"319-329","source":"Crossref","is-referenced-by-count":3,"title":["On integral bases in relative quadratic extensions"],"prefix":"10.1090","volume":"65","author":[{"given":"M.","family":"Daberkow","sequence":"first","affiliation":[]},{"given":"M.","family":"Pohst","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[1996]]},"reference":[{"key":"1","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4614-6294-1","volume-title":"The collected papers of Emil Artin","author":"Artin, Emil","year":"1965"},{"key":"2","series-title":"London Mathematical Society Student Texts","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781139171885","volume-title":"Local fields","volume":"3","author":"Cassels, J. W. S.","year":"1986","ISBN":"https:\/\/id.crossref.org\/isbn\/0521304849"},{"key":"3","unstructured":"Fachgruppe Computeralgebra der GI, Computeralgebra in Deutschland, Fachgruppe Computeralgebra der GI (1993), 212 \u2013 218."},{"key":"4","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":"5","doi-asserted-by":"publisher","first-page":"18","DOI":"10.1007\/BF01180469","article-title":"Discriminants of algebraic number fields","volume":"74","author":"Fr\u00f6lich, Albrecht","year":"1960","journal-title":"Math. Z.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5874","issn-type":"print"},{"key":"6","unstructured":"H. Hasse, Bericht \u00fcber neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlk\u00f6rper, Jahresber. Deutsch. Math.-Verein. 35 (1926)."},{"key":"7","series-title":"Graduate Texts in Mathematics","isbn-type":"print","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4757-4092-9","volume-title":"Lectures on the theory of algebraic numbers","volume":"77","author":"Hecke, Erich","year":"1981","ISBN":"https:\/\/id.crossref.org\/isbn\/0387905952"},{"key":"8","doi-asserted-by":"crossref","unstructured":"D. Hilbert, \u00dcber die Theorie des relativquadratischen Zahlk\u00f6rpers, Math. Ann. 51 (1898).","DOI":"10.1007\/BF01905120"},{"key":"9","isbn-type":"print","volume-title":"Elementary and analytic theory of algebraic numbers","author":"Narkiewicz, W\u0142adys\u0142aw","year":"1990","ISBN":"https:\/\/id.crossref.org\/isbn\/3540512500","edition":"2"},{"key":"10","unstructured":"J. Sommer, Vorlesungen \u00fcber Zahlentheorie, Teubner, Leipzig, 1907."},{"key":"11","first-page":"90","article-title":"Ein Algorithmus zur Berechnung einer Minimalbasis \u00fcber gegebener Ordnung","author":"Zassenhaus, Hans","year":"1967"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/1996-65-213\/S0025-5718-96-00686-2\/S0025-5718-96-00686-2.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/1996-65-213\/S0025-5718-96-00686-2\/S0025-5718-96-00686-2.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T21:02:19Z","timestamp":1776718939000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/1996-65-213\/S0025-5718-96-00686-2\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1996]]},"references-count":11,"journal-issue":{"issue":"213","published-print":{"date-parts":[[1996,1]]}},"alternative-id":["S0025-5718-96-00686-2"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-96-00686-2","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]]}}}