{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T22:38:04Z","timestamp":1776724684163,"version":"3.51.2"},"reference-count":11,"publisher":"American Mathematical Society (AMS)","issue":"221","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    In this paper, criteria of divisibility of the class number\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"h Superscript plus\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>h<\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">h^+<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    of the real cyclotomic field\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"bold upper Q left-parenthesis zeta Subscript p Baseline plus zeta Subscript p Superscript negative 1 Baseline right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"bold\">Q<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>\n                                \u03b6\n                                \n                              <\/mml:mi>\n                              <mml:mi>p<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:msubsup>\n                              <mml:mi>\n                                \u03b6\n                                \n                              <\/mml:mi>\n                              <mml:mi>p<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mo>\n                                  \u2212\n                                  \n                                <\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msubsup>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathbf {Q}(\\zeta _p+\\zeta _p^{-1})<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    of a prime conductor\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p\">\n                        <mml:semantics>\n                          <mml:mi>p<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">p<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and of a prime degree\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"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                    by primes\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"q\">\n                        <mml:semantics>\n                          <mml:mi>q<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">q<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    the order modulo\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"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                    of which is\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"StartFraction l minus 1 Over 2 EndFraction\">\n                        <mml:semantics>\n                          <mml:mfrac>\n                            <mml:mrow>\n                              <mml:mi>l<\/mml:mi>\n                              <mml:mo>\n                                \u2212\n                                \n                              <\/mml:mo>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:mrow>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:mfrac>\n                          <mml:annotation encoding=\"application\/x-tex\">\\frac {l-1}{2}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , are given. A corollary of these criteria is the possibility to make a computational proof that a given\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"q\">\n                        <mml:semantics>\n                          <mml:mi>q<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">q<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    does not divide\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"h Superscript plus\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>h<\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">h^+<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    for any\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p\">\n                        <mml:semantics>\n                          <mml:mi>p<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">p<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    (conductor) such that both\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"StartFraction p minus 1 Over 2 EndFraction comma StartFraction p minus 3 Over 4 EndFraction\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mfrac>\n                              <mml:mrow>\n                                <mml:mi>p<\/mml:mi>\n                                <mml:mo>\n                                  \u2212\n                                  \n                                <\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:mfrac>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mfrac>\n                              <mml:mrow>\n                                <mml:mi>p<\/mml:mi>\n                                <mml:mo>\n                                  \u2212\n                                  \n                                <\/mml:mo>\n                                <mml:mn>3<\/mml:mn>\n                              <\/mml:mrow>\n                              <mml:mn>4<\/mml:mn>\n                            <\/mml:mfrac>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\frac {p-1}{2},\\frac {p-3}{4}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    are primes. Note that on the basis of Schinzel\u2019s hypothesis there are infinitely many such primes\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p\">\n                        <mml:semantics>\n                          <mml:mi>p<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">p<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    .\n                  <\/p>","DOI":"10.1090\/s0025-5718-98-00916-8","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:44Z","timestamp":1027707284000},"page":"369-398","source":"Crossref","is-referenced-by-count":8,"title":["On divisibility of the class number \u210e\u207a of the real cyclotomic fields of prime degree \ud835\udc59"],"prefix":"10.1090","volume":"67","author":[{"given":"Stanislav","family":"Jakubec","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[1998]]},"reference":[{"issue":"1","key":"1","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/0022-314X(78)90002-1","article-title":"Computing the number of totally positive circular units which are squares","volume":"10","author":"Davis, Daniel","year":"1978","journal-title":"J. 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