{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T04:25:50Z","timestamp":1776831950059,"version":"3.51.2"},"reference-count":13,"publisher":"American Mathematical Society (AMS)","issue":"256","license":[{"start":{"date-parts":[[2007,7,11]],"date-time":"2007-07-11T00:00:00Z","timestamp":1184112000000},"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                    For a positive integer\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"m\">\n                        <mml:semantics>\n                          <mml:mi>m<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">m<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , set\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"zeta Subscript m Baseline equals exp left-parenthesis 2 pi i slash m right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>\n                                \u03b6\n                                \n                              <\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>m<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mi>exp<\/mml:mi>\n                            <mml:mo>\n                              \u2061\n                              \n                            <\/mml:mo>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mi>\n                              \u03c0\n                              \n                            <\/mml:mi>\n                            <mml:mi>i<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\/<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mi>m<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\zeta _{m}=\\exp (2\\pi i\/m)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and let\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"bold upper Z Subscript m Superscript asterisk\">\n                        <mml:semantics>\n                          <mml:msubsup>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mtext mathvariant=\"bold\">Z<\/mml:mtext>\n                            <\/mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi>m<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\n                                \u2217\n                                \n                              <\/mml:mo>\n                            <\/mml:mrow>\n                          <\/mml:msubsup>\n                          <mml:annotation encoding=\"application\/x-tex\">\\textbf {Z}_{m}^{*}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    denote the group of reduced residues modulo\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"m\">\n                        <mml:semantics>\n                          <mml:mi>m<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">m<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . Fix a congruence group\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper H\">\n                        <mml:semantics>\n                          <mml:mi>H<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">H<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    of conductor\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"m\">\n                        <mml:semantics>\n                          <mml:mi>m<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">m<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and of order\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"f\">\n                        <mml:semantics>\n                          <mml:mi>f<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">f<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . Choose integers\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"t 1 comma ellipsis comma t Subscript e Baseline\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>t<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mo>\n                              \u2026\n                              \n                            <\/mml:mo>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>t<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>e<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">t_{1},\\dots ,t_{e}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    to represent the\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"e equals phi left-parenthesis m right-parenthesis slash f\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>e<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mi>\n                              \u03d5\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>m<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\/<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mi>f<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">e=\\phi (m)\/f<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    cosets of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper H\">\n                        <mml:semantics>\n                          <mml:mi>H<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">H<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"bold upper Z Subscript m Superscript asterisk\">\n                        <mml:semantics>\n                          <mml:msubsup>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mtext mathvariant=\"bold\">Z<\/mml:mtext>\n                            <\/mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi>m<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\n                                \u2217\n                                \n                              <\/mml:mo>\n                            <\/mml:mrow>\n                          <\/mml:msubsup>\n                          <mml:annotation encoding=\"application\/x-tex\">\\textbf {Z}_{m}^{*}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . The Gauss periods\n                    <disp-formula content-type=\"math\/mathml\">\n                      \\[\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"StartLayout 1st Row  theta Subscript j Baseline equals sigma-summation Underscript x element-of upper H Endscripts zeta Subscript m Superscript t Super Subscript j Superscript x Baseline left-parenthesis 1 less-than-or-equal-to j less-than-or-equal-to e right-parenthesis EndLayout\">\n                        <mml:semantics>\n                          <mml:mtable rowspacing=\".5em\" columnspacing=\"1em\" displaystyle=\"true\">\n                            <mml:mtr>\n                              <mml:mtd>\n                                <mml:msub>\n                                  <mml:mi>\n                                    \u03b8\n                                    \n                                  <\/mml:mi>\n                                  <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                    <mml:mi>j<\/mml:mi>\n                                  <\/mml:mrow>\n                                <\/mml:msub>\n                                <mml:mo>=<\/mml:mo>\n                                <mml:munder>\n                                  <mml:mo>\n                                    \u2211\n                                    \n                                  <\/mml:mo>\n                                  <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                    <mml:mi>x<\/mml:mi>\n                                    <mml:mo>\n                                      \u2208\n                                      \n                                    <\/mml:mo>\n                                    <mml:mi>H<\/mml:mi>\n                                  <\/mml:mrow>\n                                <\/mml:munder>\n                                <mml:msubsup>\n                                  <mml:mi>\n                                    \u03b6\n                                    \n                                  <\/mml:mi>\n                                  <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                    <mml:mi>m<\/mml:mi>\n                                  <\/mml:mrow>\n                                  <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                    <mml:msub>\n                                      <mml:mi>t<\/mml:mi>\n                                      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                        <mml:mi>j<\/mml:mi>\n                                      <\/mml:mrow>\n                                    <\/mml:msub>\n                                    <mml:mi>x<\/mml:mi>\n                                  <\/mml:mrow>\n                                <\/mml:msubsup>\n                                <mml:mspace width=\"thickmathspace\"\/>\n                                <mml:mspace width=\"thickmathspace\"\/>\n                                <mml:mspace width=\"thickmathspace\"\/>\n                                <mml:mo stretchy=\"false\">(<\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                                <mml:mo>\n                                  \u2264\n                                  \n                                <\/mml:mo>\n                                <mml:mi>j<\/mml:mi>\n                                <mml:mo>\n                                  \u2264\n                                  \n                                <\/mml:mo>\n                                <mml:mi>e<\/mml:mi>\n                                <mml:mo stretchy=\"false\">)<\/mml:mo>\n                              <\/mml:mtd>\n                            <\/mml:mtr>\n                          <\/mml:mtable>\n                          <mml:annotation encoding=\"application\/x-tex\">\\displaylines { \\theta _{j} =\\sum _{x \\in H} \\zeta _{m}^{t_{j}x} \\;\\;\\; (1 \\leq j \\leq e) }<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                      \\]\n                    <\/disp-formula>\n                    corresponding to\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper H\">\n                        <mml:semantics>\n                          <mml:mi>H<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">H<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    are conjugate and distinct over\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"bold upper Q\">\n                        <mml:semantics>\n                          <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                            <mml:mtext mathvariant=\"bold\">Q<\/mml:mtext>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\textbf {Q}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    with minimal polynomial\n                    <disp-formula content-type=\"math\/mathml\">\n                      \\[\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"StartLayout 1st Row  g left-parenthesis x right-parenthesis equals x Superscript e Baseline plus c 1 x Superscript e minus 1 Baseline plus midline-horizontal-ellipsis plus c Subscript e minus 1 Baseline x plus c Subscript e Baseline period EndLayout\">\n                        <mml:semantics>\n                          <mml:mtable rowspacing=\".5em\" columnspacing=\"1em\" displaystyle=\"true\">\n                            <mml:mtr>\n                              <mml:mtd>\n                                <mml:mi>g<\/mml:mi>\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:mi>e<\/mml:mi>\n                                  <\/mml:mrow>\n                                <\/mml:msup>\n                                <mml:mo>+<\/mml:mo>\n                                <mml:msub>\n                                  <mml:mi>c<\/mml:mi>\n                                  <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                    <mml:mn>1<\/mml:mn>\n                                  <\/mml:mrow>\n                                <\/mml:msub>\n                                <mml:msup>\n                                  <mml:mi>x<\/mml:mi>\n                                  <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                    <mml:mi>e<\/mml:mi>\n                                    <mml:mo>\n                                      \u2212\n                                      \n                                    <\/mml:mo>\n                                    <mml:mn>1<\/mml:mn>\n                                  <\/mml:mrow>\n                                <\/mml:msup>\n                                <mml:mo>+<\/mml:mo>\n                                <mml:mo>\n                                  \u22ef\n                                  \n                                <\/mml:mo>\n                                <mml:mo>+<\/mml:mo>\n                                <mml:msub>\n                                  <mml:mi>c<\/mml:mi>\n                                  <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                    <mml:mi>e<\/mml:mi>\n                                    <mml:mo>\n                                      \u2212\n                                      \n                                    <\/mml:mo>\n                                    <mml:mn>1<\/mml:mn>\n                                  <\/mml:mrow>\n                                <\/mml:msub>\n                                <mml:mi>x<\/mml:mi>\n                                <mml:mo>+<\/mml:mo>\n                                <mml:msub>\n                                  <mml:mi>c<\/mml:mi>\n                                  <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                    <mml:mi>e<\/mml:mi>\n                                  <\/mml:mrow>\n                                <\/mml:msub>\n                                <mml:mo>.<\/mml:mo>\n                              <\/mml:mtd>\n                            <\/mml:mtr>\n                          <\/mml:mtable>\n                          <mml:annotation encoding=\"application\/x-tex\">\\displaylines { g(x) = x^{e} + c_{1}x^{e-1} + \\cdots + c_{e-1} x + c_{e}. }<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                      \\]\n                    <\/disp-formula>\n                    To determine the coefficients of the period polynomial\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"g left-parenthesis x right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>g<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">g(x)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    (or equivalently, its reciprocal polynomial\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper G left-parenthesis upper X right-parenthesis equals upper X Superscript e Baseline g left-parenthesis upper X Superscript negative 1 Baseline right-parenthesis right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>G<\/mml:mi>\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:mi>e<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mi>g<\/mml:mi>\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:mo>\n                                  \u2212\n                                  \n                                <\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">G(X)=X^{e}g(X^{-1}))<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is a classical problem dating back to Gauss. Previous work of the author, and Gupta and Zagier, primarily treated the case\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"m equals p\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>m<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mi>p<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">m=p<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , an odd prime, with\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"f greater-than 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>f<\/mml:mi>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">f &gt;1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    fixed. In this setting, it is known for certain integral power series\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper A left-parenthesis upper X right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>A<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>X<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">A(X)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper B left-parenthesis upper X right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>B<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>X<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">B(X)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , that for any positive integer\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper N\">\n                        <mml:semantics>\n                          <mml:mi>N<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">N<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    <disp-formula content-type=\"math\/mathml\">\n                      \\[\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"StartLayout 1st Row  upper G left-parenthesis upper X right-parenthesis identical-to upper A left-parenthesis upper X right-parenthesis dot upper B left-parenthesis upper X right-parenthesis Superscript StartFraction p minus 1 Over f EndFraction Baseline left-parenthesis mod upper X Superscript upper N Baseline right-parenthesis EndLayout\">\n                        <mml:semantics>\n                          <mml:mtable rowspacing=\".5em\" columnspacing=\"1em\" displaystyle=\"true\">\n                            <mml:mtr>\n                              <mml:mtd>\n                                <mml:mi>G<\/mml:mi>\n                                <mml:mo stretchy=\"false\">(<\/mml:mo>\n                                <mml:mi>X<\/mml:mi>\n                                <mml:mo stretchy=\"false\">)<\/mml:mo>\n                                <mml:mo>\n                                  \u2261\n                                  \n                                <\/mml:mo>\n                                <mml:mi>A<\/mml:mi>\n                                <mml:mo stretchy=\"false\">(<\/mml:mo>\n                                <mml:mi>X<\/mml:mi>\n                                <mml:mo stretchy=\"false\">)<\/mml:mo>\n                                <mml:mo>\n                                  \u22c5\n                                  \n                                <\/mml:mo>\n                                <mml:mi>B<\/mml:mi>\n                                <mml:mo stretchy=\"false\">(<\/mml:mo>\n                                <mml:mi>X<\/mml:mi>\n                                <mml:msup>\n                                  <mml:mo stretchy=\"false\">)<\/mml:mo>\n                                  <mml:mrow class=\"MJX-TeXAtom-ORD\">\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:mi>f<\/mml:mi>\n                                    <\/mml:mfrac>\n                                  <\/mml:mrow>\n                                <\/mml:msup>\n                                <mml:mspace width=\"thickmathspace\"\/>\n                                <mml:mspace width=\"thickmathspace\"\/>\n                                <mml:mspace width=\"thickmathspace\"\/>\n                                <mml:mo stretchy=\"false\">(<\/mml:mo>\n                                <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                  <mml:mtext>mod<\/mml:mtext>\n                                <\/mml:mrow>\n                                <mml:mspace width=\"thickmathspace\"\/>\n                                <mml:msup>\n                                  <mml:mi>X<\/mml:mi>\n                                  <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                    <mml:mi>N<\/mml:mi>\n                                  <\/mml:mrow>\n                                <\/mml:msup>\n                                <mml:mo stretchy=\"false\">)<\/mml:mo>\n                              <\/mml:mtd>\n                            <\/mml:mtr>\n                          <\/mml:mtable>\n                          <mml:annotation encoding=\"application\/x-tex\">\\displaylines { G(X) \\equiv A(X)\\cdot B(X)^{\\frac {p-1}{f}} \\;\\;\\;(\\textrm {mod}\\;X^{N}) }<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                      \\]\n                    <\/disp-formula>\n                    holds in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"bold upper Z left-bracket upper X right-bracket\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mtext mathvariant=\"bold\">Z<\/mml:mtext>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">[<\/mml:mo>\n                            <mml:mi>X<\/mml:mi>\n                            <mml:mo stretchy=\"false\">]<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\textbf {Z}[X]<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    for all primes\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p identical-to 1 left-parenthesis mod f right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>p<\/mml:mi>\n                            <mml:mo>\n                              \u2261\n                              \n                            <\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mtext>mod<\/mml:mtext>\n                            <\/mml:mrow>\n                            <mml:mspace width=\"thickmathspace\"\/>\n                            <mml:mi>f<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">p \\equiv 1(\\textrm {mod}\\; f)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    except those in an effectively determinable finite set. Here we describe an analogous result for the case\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"m equals p Superscript alpha\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>m<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>p<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>\n                                  \u03b1\n                                  \n                                <\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">m=p^{\\alpha }<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , a prime power (\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"alpha greater-than 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03b1\n                              \n                            <\/mml:mi>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\alpha &gt; 1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ). The methods extend for odd prime powers\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p Superscript alpha\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>p<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi>\n                                \u03b1\n                                \n                              <\/mml:mi>\n                            <\/mml:mrow>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">p^{\\alpha }<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    to give a similar result for certain twisted Gauss periods of the form\n                    <disp-formula content-type=\"math\/mathml\">\n                      \\[\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"StartLayout 1st Row  psi Subscript j Baseline equals i Superscript asterisk Baseline StartRoot p EndRoot sigma-summation Underscript x element-of upper H Endscripts left-parenthesis StartFraction t Subscript j Baseline x Over p EndFraction right-parenthesis zeta Subscript p Sub Superscript alpha Subscript Superscript t Super Subscript j Superscript x Baseline left-parenthesis 1 less-than-or-equal-to j less-than-or-equal-to e right-parenthesis comma EndLayout\">\n                        <mml:semantics>\n                          <mml:mtable rowspacing=\".5em\" columnspacing=\"1em\" displaystyle=\"true\">\n                            <mml:mtr>\n                              <mml:mtd>\n                                <mml:msub>\n                                  <mml:mi>\n                                    \u03c8\n                                    \n                                  <\/mml:mi>\n                                  <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                    <mml:mi>j<\/mml:mi>\n                                  <\/mml:mrow>\n                                <\/mml:msub>\n                                <mml:mo>=<\/mml:mo>\n                                <mml:msup>\n                                  <mml:mi>i<\/mml:mi>\n                                  <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                    <mml:mo>\n                                      \u2217\n                                      \n                                    <\/mml:mo>\n                                  <\/mml:mrow>\n                                <\/mml:msup>\n                                <mml:msqrt>\n                                  <mml:mi>p<\/mml:mi>\n                                <\/mml:msqrt>\n                                <mml:munder>\n                                  <mml:mo>\n                                    \u2211\n                                    \n                                  <\/mml:mo>\n                                  <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                    <mml:mi>x<\/mml:mi>\n                                    <mml:mo>\n                                      \u2208\n                                      \n                                    <\/mml:mo>\n                                    <mml:mi>H<\/mml:mi>\n                                  <\/mml:mrow>\n                                <\/mml:munder>\n                                <mml:mo stretchy=\"false\">(<\/mml:mo>\n                                <mml:mfrac>\n                                  <mml:mrow>\n                                    <mml:msub>\n                                      <mml:mi>t<\/mml:mi>\n                                      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                        <mml:mi>j<\/mml:mi>\n                                      <\/mml:mrow>\n                                    <\/mml:msub>\n                                    <mml:mi>x<\/mml:mi>\n                                  <\/mml:mrow>\n                                  <mml:mi>p<\/mml:mi>\n                                <\/mml:mfrac>\n                                <mml:mo stretchy=\"false\">)<\/mml:mo>\n                                <mml:msubsup>\n                                  <mml:mi>\n                                    \u03b6\n                                    \n                                  <\/mml:mi>\n                                  <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                    <mml:msup>\n                                      <mml:mi>p<\/mml:mi>\n                                      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                        <mml:mi>\n                                          \u03b1\n                                          \n                                        <\/mml:mi>\n                                      <\/mml:mrow>\n                                    <\/mml:msup>\n                                  <\/mml:mrow>\n                                  <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                    <mml:msub>\n                                      <mml:mi>t<\/mml:mi>\n                                      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                        <mml:mi>j<\/mml:mi>\n                                      <\/mml:mrow>\n                                    <\/mml:msub>\n                                    <mml:mi>x<\/mml:mi>\n                                  <\/mml:mrow>\n                                <\/mml:msubsup>\n                                <mml:mspace width=\"thickmathspace\"\/>\n                                <mml:mspace width=\"thickmathspace\"\/>\n                                <mml:mo stretchy=\"false\">(<\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                                <mml:mo>\n                                  \u2264\n                                  \n                                <\/mml:mo>\n                                <mml:mi>j<\/mml:mi>\n                                <mml:mo>\n                                  \u2264\n                                  \n                                <\/mml:mo>\n                                <mml:mi>e<\/mml:mi>\n                                <mml:mo stretchy=\"false\">)<\/mml:mo>\n                                <mml:mo>,<\/mml:mo>\n                              <\/mml:mtd>\n                            <\/mml:mtr>\n                          <\/mml:mtable>\n                          <mml:annotation encoding=\"application\/x-tex\">\\displaylines { \\psi _{j} = i^{*} \\sqrt {p} \\sum _{x \\in H} (\\frac {t_{j}x}{p}) \\zeta _{p^{\\alpha }}^{t_{j}x} \\;\\;(1 \\leq j \\leq e),}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                      \\]\n                    <\/disp-formula>\n                    where\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-parenthesis StartFraction Over p EndFraction right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mfrac>\n                              <mml:mrow\/>\n                              <mml:mi>p<\/mml:mi>\n                            <\/mml:mfrac>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">(\\frac { }{p})<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    denotes the usual Legendre symbol and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"i Superscript asterisk Baseline equals i Superscript StartFraction left-parenthesis p minus 1 right-parenthesis squared Over 4 EndFraction\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msup>\n                              <mml:mi>i<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mo>\n                                  \u2217\n                                  \n                                <\/mml:mo>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>i<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mfrac>\n                                  <mml:mrow>\n                                    <mml:mo stretchy=\"false\">(<\/mml:mo>\n                                    <mml:mi>p<\/mml:mi>\n                                    <mml:mo>\n                                      \u2212\n                                      \n                                    <\/mml:mo>\n                                    <mml:mn>1<\/mml:mn>\n                                    <mml:msup>\n                                      <mml:mo stretchy=\"false\">)<\/mml:mo>\n                                      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                        <mml:mn>2<\/mml:mn>\n                                      <\/mml:mrow>\n                                    <\/mml:msup>\n                                  <\/mml:mrow>\n                                  <mml:mn>4<\/mml:mn>\n                                <\/mml:mfrac>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">i^{*}= i^{\\frac {(p-1)^{2}}{4}}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    .\n                  <\/p>","DOI":"10.1090\/s0025-5718-06-01885-0","type":"journal-article","created":{"date-parts":[[2006,8,16]],"date-time":"2006-08-16T10:28:45Z","timestamp":1155724125000},"page":"2021-2035","source":"Crossref","is-referenced-by-count":5,"title":["On the minimal polynomial of Gauss periods for prime powers"],"prefix":"10.1090","volume":"75","author":[{"given":"S.","family":"Gurak","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2006,7,11]]},"reference":[{"key":"1","series-title":"Canadian Mathematical Society Series of Monographs and Advanced Texts","isbn-type":"print","volume-title":"Gauss and Jacobi sums","author":"Berndt, Bruce C.","year":"1998","ISBN":"https:\/\/id.crossref.org\/isbn\/0471128074"},{"key":"2","series-title":"Pure and Applied Mathematics, Vol. 20","volume-title":"Number theory","author":"Borevich, A. I.","year":"1966"},{"issue":"203","key":"3","doi-asserted-by":"publisher","first-page":"131","DOI":"10.2307\/2152941","article-title":"On computing factors of cyclotomic polynomials","volume":"61","author":"Brent, Richard P.","year":"1993","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"4","unstructured":"L.E. Dickson, Elementary Theory of Equations, Wiley, New York."},{"key":"5","volume-title":"Disquisitiones arithmeticae","author":"Gauss, Carl Friedrich","year":"1966"},{"issue":"201","key":"6","doi-asserted-by":"publisher","first-page":"385","DOI":"10.2307\/2153175","article-title":"On the coefficients of the minimal polynomials of Gaussian periods","volume":"60","author":"Gupta, S.","year":"1993","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"2","key":"7","doi-asserted-by":"crossref","first-page":"347","DOI":"10.2140\/pjm.1982.102.347","article-title":"Minimal polynomials for Gauss circulants and cyclotomic units","volume":"102","author":"Gurak, S.","year":"1982","journal-title":"Pacific J. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0030-8730","issn-type":"print"},{"issue":"2","key":"8","doi-asserted-by":"crossref","first-page":"313","DOI":"10.2140\/pjm.1984.112.313","article-title":"Minimal polynomials for circular numbers","volume":"112","author":"Gurak, S.","year":"1984","journal-title":"Pacific J. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0030-8730","issn-type":"print"},{"key":"9","doi-asserted-by":"crossref","unstructured":"S. Gurak, \"Minimal polynomials for Gauss periods with f=2,\" Acta Arith. 121 (2006), 233\u2013257.","DOI":"10.4064\/aa121-3-3"},{"key":"10","unstructured":"S. Gurak, \"Explicit evaluation of multi-dimensional Kloosterman sums for prime powers\" (to appear)."},{"key":"11","series-title":"Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Ber\\\"{u}cksichtigung der Anwendungsgebiete, Band LIX","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-52795-1","volume-title":"Vorlesungen \\\"{u}ber Zahlentheorie","author":"Hasse, Helmut","year":"1950"},{"key":"12","series-title":"Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-82465-4","volume-title":"Class field theory","volume":"280","author":"Neukirch, J\u00fcrgen","year":"1986","ISBN":"https:\/\/id.crossref.org\/isbn\/3540152512"},{"issue":"1","key":"13","doi-asserted-by":"publisher","first-page":"91","DOI":"10.1007\/BF01180579","article-title":"\u00dcber die Kloostermanschen Summen \ud835\udc46(\ud835\udc62,\ud835\udc63;\ud835\udc5e)","volume":"34","author":"Sali\u00e9, Hans","year":"1932","journal-title":"Math. Z.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5874","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2006-75-256\/S0025-5718-06-01885-0\/S0025-5718-06-01885-0.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2006-75-256\/S0025-5718-06-01885-0\/S0025-5718-06-01885-0.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T14:41:34Z","timestamp":1776782494000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2006-75-256\/S0025-5718-06-01885-0\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2006,7,11]]},"references-count":13,"journal-issue":{"issue":"256","published-print":{"date-parts":[[2006,10]]}},"alternative-id":["S0025-5718-06-01885-0"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-06-01885-0","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":[[2006,7,11]]}}}