{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T10:10:01Z","timestamp":1776766201570,"version":"3.51.2"},"reference-count":11,"publisher":"American Mathematical Society (AMS)","issue":"220","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    We consider the totally real cyclic quintic fields\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper K Subscript n Baseline equals double-struck upper Q left-parenthesis theta Subscript n Baseline right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>K<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>n<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo>=<\/mml:mo>\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:msub>\n                              <mml:mi>\n                                \u03d1\n                                \n                              <\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>n<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">K_{n}=\\mathbb {Q}(\\vartheta _{n})<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , generated by a root\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"theta Subscript n\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>\n                              \u03d1\n                              \n                            <\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi>n<\/mml:mi>\n                            <\/mml:mrow>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">\\vartheta _{n}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    of the polynomial\n                    <disp-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"StartLayout 1st Row  f Subscript n Baseline left-parenthesis x right-parenthesis equals x Superscript 5 Baseline plus n squared x Superscript 4 Baseline minus left-parenthesis 2 n cubed plus 6 n squared plus 10 n plus 10 right-parenthesis x cubed plus left-parenthesis n Superscript 4 Baseline plus 5 n cubed plus 11 n squared plus 15 n plus 5 right-parenthesis x squared plus left-parenthesis n cubed plus 4 n squared plus 10 n plus 10 right-parenthesis x plus 1 period EndLayout\">\n                        <mml:semantics>\n                          <mml:mtable columnwidth=\"100%\" width=\"85%\" rowspacing=\".5em\" side=\"left\" displaystyle=\"true\">\n                            <mml:mtr>\n                              <mml:mtd columnalign=\"left\">\n                                <mml:msub>\n                                  <mml:mi>f<\/mml:mi>\n                                  <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                    <mml:mi>n<\/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:msup>\n                                  <mml:mi>x<\/mml:mi>\n                                  <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                    <mml:mn>5<\/mml:mn>\n                                  <\/mml:mrow>\n                                <\/mml:msup>\n                                <mml:mo>+<\/mml:mo>\n                                <mml:msup>\n                                  <mml:mi>n<\/mml:mi>\n                                  <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                    <mml:mn>2<\/mml:mn>\n                                  <\/mml:mrow>\n                                <\/mml:msup>\n                                <mml:msup>\n                                  <mml:mi>x<\/mml:mi>\n                                  <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                    <mml:mn>4<\/mml:mn>\n                                  <\/mml:mrow>\n                                <\/mml:msup>\n                                <mml:mo>\n                                  \u2212\n                                  \n                                <\/mml:mo>\n                                <mml:mo stretchy=\"false\">(<\/mml:mo>\n                                <mml:mn>2<\/mml:mn>\n                                <mml:msup>\n                                  <mml:mi>n<\/mml:mi>\n                                  <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                    <mml:mn>3<\/mml:mn>\n                                  <\/mml:mrow>\n                                <\/mml:msup>\n                                <mml:mo>+<\/mml:mo>\n                                <mml:mn>6<\/mml:mn>\n                                <mml:msup>\n                                  <mml:mi>n<\/mml:mi>\n                                  <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                    <mml:mn>2<\/mml:mn>\n                                  <\/mml:mrow>\n                                <\/mml:msup>\n                                <mml:mo>+<\/mml:mo>\n                                <mml:mn>10<\/mml:mn>\n                                <mml:mi>n<\/mml:mi>\n                                <mml:mo>+<\/mml:mo>\n                                <mml:mn>10<\/mml:mn>\n                                <mml:mo stretchy=\"false\">)<\/mml:mo>\n                                <mml:msup>\n                                  <mml:mi>x<\/mml:mi>\n                                  <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                    <mml:mn>3<\/mml:mn>\n                                  <\/mml:mrow>\n                                <\/mml:msup>\n                                <mml:mtext>\u00a0<\/mml:mtext>\n                                <mml:mo>+<\/mml:mo>\n                                <mml:mo stretchy=\"false\">(<\/mml:mo>\n                                <mml:msup>\n                                  <mml:mi>n<\/mml:mi>\n                                  <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                    <mml:mn>4<\/mml:mn>\n                                  <\/mml:mrow>\n                                <\/mml:msup>\n                                <mml:mo>+<\/mml:mo>\n                                <mml:mn>5<\/mml:mn>\n                                <mml:msup>\n                                  <mml:mi>n<\/mml:mi>\n                                  <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                    <mml:mn>3<\/mml:mn>\n                                  <\/mml:mrow>\n                                <\/mml:msup>\n                                <mml:mo>+<\/mml:mo>\n                                <mml:mn>11<\/mml:mn>\n                                <mml:msup>\n                                  <mml:mi>n<\/mml:mi>\n                                  <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                    <mml:mn>2<\/mml:mn>\n                                  <\/mml:mrow>\n                                <\/mml:msup>\n                                <mml:mo>+<\/mml:mo>\n                                <mml:mn>15<\/mml:mn>\n                                <mml:mi>n<\/mml:mi>\n                                <mml:mo>+<\/mml:mo>\n                                <mml:mn>5<\/mml:mn>\n                                <mml:mo stretchy=\"false\">)<\/mml:mo>\n                                <mml:msup>\n                                  <mml:mi>x<\/mml:mi>\n                                  <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                    <mml:mn>2<\/mml:mn>\n                                  <\/mml:mrow>\n                                <\/mml:msup>\n                                <mml:mo>+<\/mml:mo>\n                                <mml:mo stretchy=\"false\">(<\/mml:mo>\n                                <mml:msup>\n                                  <mml:mi>n<\/mml:mi>\n                                  <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                    <mml:mn>3<\/mml:mn>\n                                  <\/mml:mrow>\n                                <\/mml:msup>\n                                <mml:mo>+<\/mml:mo>\n                                <mml:mn>4<\/mml:mn>\n                                <mml:msup>\n                                  <mml:mi>n<\/mml:mi>\n                                  <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                    <mml:mn>2<\/mml:mn>\n                                  <\/mml:mrow>\n                                <\/mml:msup>\n                                <mml:mo>+<\/mml:mo>\n                                <mml:mn>10<\/mml:mn>\n                                <mml:mi>n<\/mml:mi>\n                                <mml:mo>+<\/mml:mo>\n                                <mml:mn>10<\/mml:mn>\n                                <mml:mo stretchy=\"false\">)<\/mml:mo>\n                                <mml:mi>x<\/mml:mi>\n                                <mml:mo>+<\/mml:mo>\n                                <mml:mn>1.<\/mml:mn>\n                              <\/mml:mtd>\n                            <\/mml:mtr>\n                          <\/mml:mtable>\n                          <mml:annotation encoding=\"application\/x-tex\">\\begin{multline*} f_{n}(x)=x^{5}+n^{2}x^{4}-(2n^{3}+6n^{2}+10n+10)x^{3}\\ +(n^{4}+5n^{3}+11n^{2}+15n+5)x^{2}+(n^{3}+4n^{2}+10n+10)x+1. \\end{multline*}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/disp-formula>\n                    Assuming that\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"m equals n Superscript 4 Baseline plus 5 n cubed plus 15 n squared plus 25 n plus 25\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>m<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>n<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>4<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mn>5<\/mml:mn>\n                            <mml:msup>\n                              <mml:mi>n<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>3<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mn>15<\/mml:mn>\n                            <mml:msup>\n                              <mml:mi>n<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>2<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mn>25<\/mml:mn>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mn>25<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">m=n^{4}+5n^{3}+15n^{2}+25n+25<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is square free, we compute explicitly an integral basis and a set of fundamental units of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper K Subscript n\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>K<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi>n<\/mml:mi>\n                            <\/mml:mrow>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">K_{n}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and prove that\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper K Subscript n\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>K<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi>n<\/mml:mi>\n                            <\/mml:mrow>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">K_{n}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    has a power integral basis only for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n equals negative 1 comma negative 2\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">n=-1,-2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . For\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n equals negative 1 comma negative 2\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">n=-1,-2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    (both values presenting the same field) all generators of power integral bases are computed.\n                  <\/p>","DOI":"10.1090\/s0025-5718-97-00868-5","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:44Z","timestamp":1027707284000},"page":"1689-1696","source":"Crossref","is-referenced-by-count":14,"title":["Power integral bases in a parametric family of totally real cyclic quintics"],"prefix":"10.1090","volume":"66","author":[{"given":"Istv\u00e1n","family":"Ga\u00e1l","sequence":"first","affiliation":[]},{"given":"Michael","family":"Pohst","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[1997]]},"reference":[{"key":"1","unstructured":"M.Daberkow, C.Fieker, J.Kl\u00fcners, M.Pohst, K.Roegner, M.Sch\u00f6rnig & K.Wildanger, Kant V4, J. 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