{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T20:56:26Z","timestamp":1776804986783,"version":"3.51.2"},"reference-count":8,"publisher":"American Mathematical Society (AMS)","issue":"318","license":[{"start":{"date-parts":[[2019,10,30]],"date-time":"2019-10-30T00:00:00Z","timestamp":1572393600000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["EP\/K034383\/1"],"award-info":[{"award-number":["EP\/K034383\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["FT160100094"],"award-info":[{"award-number":["FT160100094"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000923","name":"Australian Research Council","doi-asserted-by":"publisher","award":["EP\/K034383\/1"],"award-info":[{"award-number":["EP\/K034383\/1"]}],"id":[{"id":"10.13039\/501100000923","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000923","name":"Australian Research Council","doi-asserted-by":"publisher","award":["FT160100094"],"award-info":[{"award-number":["FT160100094"]}],"id":[{"id":"10.13039\/501100000923","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    We prove that for all\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"q greater-than 211\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>q<\/mml:mi>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mn>211<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">q&gt;211<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , there always exists a primitive root\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"g\">\n                        <mml:semantics>\n                          <mml:mi>g<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">g<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    in the finite field\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"double-struck upper F Subscript q\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"double-struck\">F<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi>q<\/mml:mi>\n                            <\/mml:mrow>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathbb {F}_{q}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    such that\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper Q left-parenthesis g right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>Q<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>g<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">Q(g)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is also a primitive root, where\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper Q left-parenthesis x right-parenthesis equals a x squared plus b x plus c\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>Q<\/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:mi>a<\/mml:mi>\n                            <mml:msup>\n                              <mml:mi>x<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mi>b<\/mml:mi>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mi>c<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">Q(x)= ax^2 + bx + c<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is a quadratic polynomial with\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"a comma b comma c element-of double-struck upper F Subscript q Baseline\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>a<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>b<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>c<\/mml:mi>\n                            <mml:mo>\n                              \u2208\n                              \n                            <\/mml:mo>\n                            <mml:msub>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi mathvariant=\"double-struck\">F<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>q<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">a, b, c\\in \\mathbb {F}_{q}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    such that\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"b squared minus 4 a c not-equals 0\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msup>\n                              <mml:mi>b<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>2<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mn>4<\/mml:mn>\n                            <mml:mi>a<\/mml:mi>\n                            <mml:mi>c<\/mml:mi>\n                            <mml:mo>\n                              \u2260\n                              \n                            <\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">b^{2} - 4ac \\neq 0<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    .\n                  <\/p>","DOI":"10.1090\/mcom\/3390","type":"journal-article","created":{"date-parts":[[2018,7,25]],"date-time":"2018-07-25T09:28:34Z","timestamp":1532510914000},"page":"1903-1912","source":"Crossref","is-referenced-by-count":18,"title":["Primitive values of quadratic polynomials in a finite field"],"prefix":"10.1090","volume":"88","author":[{"given":"Andrew","family":"Booker","sequence":"first","affiliation":[]},{"given":"Stephen","family":"Cohen","sequence":"additional","affiliation":[]},{"given":"Nicole","family":"Sutherland","sequence":"additional","affiliation":[]},{"given":"Tim","family":"Trudgian","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2018,10,30]]},"reference":[{"key":"1","doi-asserted-by":"crossref","unstructured":"G. Bailey, S. D. Cohen, N. Sutherland, and T. Trudgian, Existence results for primitive elements in cubic and quartic extensions of a finite field, Math. Comp., electronically published on May 18, 2018, DOI:10.1090\/mcom\/3357 (to appear in print).","DOI":"10.1090\/mcom\/3357"},{"key":"2","unstructured":"A. R. Booker, S. D. Cohen, N. Sutherland, and T. 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