{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T08:56:30Z","timestamp":1776848190975,"version":"3.51.2"},"reference-count":10,"publisher":"American Mathematical Society (AMS)","issue":"316","license":[{"start":{"date-parts":[[2019,5,18]],"date-time":"2019-05-18T00:00:00Z","timestamp":1558137600000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"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                    With\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:mi>q<\/mml:mi>\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                    the finite field of\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                    elements, we investigate the following question. If\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"gamma\">\n                        <mml:semantics>\n                          <mml:mi>\n                            \u03b3\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\gamma<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    generates\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 Sub Superscript n\">\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:msup>\n                                <mml:mi>q<\/mml:mi>\n                                <mml:mi>n<\/mml:mi>\n                              <\/mml:msup>\n                            <\/mml:mrow>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathbb {F}_{q^n}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    over\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:mi>q<\/mml:mi>\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                    and if\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"beta\">\n                        <mml:semantics>\n                          <mml:mi>\n                            \u03b2\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\beta<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is a nonzero element of\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 Sub Superscript n\">\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:msup>\n                                <mml:mi>q<\/mml:mi>\n                                <mml:mi>n<\/mml:mi>\n                              <\/mml:msup>\n                            <\/mml:mrow>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathbb {F}_{q^n}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , is there always an\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"a element-of double-struck upper F Subscript q\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>a<\/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:mi>q<\/mml:mi>\n                            <\/mml:msub>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">a \\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=\"beta left-parenthesis gamma plus a right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03b2\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>\n                              \u03b3\n                              \n                            <\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mi>a<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\beta (\\gamma + a)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is a primitive element? We resolve this case when\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n equals 3\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>3<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">n=3<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , thereby proving a conjecture by Cohen. We also substantially improve on what is known when\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n equals 4\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>4<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">n=4<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    .\n                  <\/p>","DOI":"10.1090\/mcom\/3357","type":"journal-article","created":{"date-parts":[[2018,2,7]],"date-time":"2018-02-07T09:22:41Z","timestamp":1517995361000},"page":"931-947","source":"Crossref","is-referenced-by-count":6,"title":["Existence results for primitive elements in cubic and quartic extensions of a finite field"],"prefix":"10.1090","volume":"88","author":[{"given":"Geoff","family":"Bailey","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":[[2018,5,18]]},"reference":[{"key":"1","unstructured":"G. Bailey, The Line Problem for Cubic Extensions: http:\/\/magma.maths.usyd.edu.au\/\u02dc geoff\/L3\/"},{"key":"2","unstructured":"W. Bosma, J. J. Cannon, C. Fieker, A. Steel (eds), Handbook of Magma Functions V2.23 (2017): http:\/\/magma.maths.usyd.edu.au\/magma\/handbook\/"},{"key":"3","doi-asserted-by":"publisher","first-page":"4","DOI":"10.1093\/qmath\/4.1.4","article-title":"Distribution of primitive roots in a finite field","volume":"4","author":"Carlitz, L.","year":"1953","journal-title":"Quart. J. Math. Oxford Ser. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0033-5606","issn-type":"print"},{"issue":"2","key":"4","doi-asserted-by":"publisher","first-page":"221","DOI":"10.1112\/jlms\/s2-27.2.221","article-title":"Primitive roots in the quadratic extension of a finite field","volume":"27","author":"Cohen, Stephen D.","year":"1983","journal-title":"J. London Math. Soc. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0024-6107","issn-type":"print"},{"issue":"3","key":"5","first-page":"189","article-title":"Generators of the cubic extension of a finite field","volume":"1","author":"Cohen, Stephen D.","year":"2009","journal-title":"J. Comb. Number Theory","ISSN":"https:\/\/id.crossref.org\/issn\/1942-5600","issn-type":"print"},{"key":"6","isbn-type":"print","doi-asserted-by":"publisher","first-page":"113","DOI":"10.1090\/conm\/518\/10200","article-title":"Primitive elements on lines in extensions of finite fields","author":"Cohen, Stephen D.","year":"2010","ISBN":"https:\/\/id.crossref.org\/isbn\/9780821847862"},{"key":"7","doi-asserted-by":"crossref","unstructured":"H. Davenport, On primitive roots in finite fields, Quart. J. 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Codes Cryptogr.","ISSN":"https:\/\/id.crossref.org\/issn\/0925-1022","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.ams.org\/mcom\/2019-88-316\/S0025-5718-2018-03357-9\/mcom3357_AM.pdf","content-type":"application\/pdf","content-version":"am","intended-application":"syndication"},{"URL":"http:\/\/www.ams.org\/mcom\/2019-88-316\/S0025-5718-2018-03357-9\/S0025-5718-2018-03357-9.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2019-88-316\/S0025-5718-2018-03357-9\/S0025-5718-2018-03357-9.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T19:59:37Z","timestamp":1776801577000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2019-88-316\/S0025-5718-2018-03357-9\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,5,18]]},"references-count":10,"journal-issue":{"issue":"316","published-print":{"date-parts":[[2019,3]]}},"alternative-id":["S0025-5718-2018-03357-9"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/3357","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":[[2018,5,18]]}}}