{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T03:11:11Z","timestamp":1776827471792,"version":"3.51.2"},"reference-count":14,"publisher":"American Mathematical Society (AMS)","issue":"236","license":[{"start":{"date-parts":[[2002,5,11]],"date-time":"2002-05-11T00:00:00Z","timestamp":1021075200000},"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>Let <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"m greater-than 2\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi>m<\/mml:mi>\n      <mml:mo>&gt;<\/mml:mo>\n      <mml:mn>2<\/mml:mn>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">m&gt; 2<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>, <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"zeta Subscript m\">\n  <mml:semantics>\n    <mml:msub>\n      <mml:mi>\u03b6<\/mml:mi>\n      <mml:mi>m<\/mml:mi>\n    <\/mml:msub>\n    <mml:annotation encoding=\"application\/x-tex\">\\zeta _m<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> an <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>-th primitive root of 1, <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"q identical-to 1\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi>q<\/mml:mi>\n      <mml:mo>\u2261<\/mml:mo>\n      <mml:mn>1<\/mml:mn>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">q\\equiv 1<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> mod <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"2 m\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mn>2<\/mml:mn>\n      <mml:mi>m<\/mml:mi>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">2m<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> a prime number, <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"s equals s Subscript q\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi>s<\/mml:mi>\n      <mml:mo>=<\/mml:mo>\n      <mml:msub>\n        <mml:mi>s<\/mml:mi>\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\">s=s_{q}<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> a primitive root modulo <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> and <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"f equals f Subscript q Baseline equals left-parenthesis q minus 1 right-parenthesis slash m\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi>f<\/mml:mi>\n      <mml:mo>=<\/mml:mo>\n      <mml:msub>\n        <mml:mi>f<\/mml:mi>\n        <mml:mrow class=\"MJX-TeXAtom-ORD\">\n          <mml:mi>q<\/mml:mi>\n        <\/mml:mrow>\n      <\/mml:msub>\n      <mml:mo>=<\/mml:mo>\n      <mml:mo stretchy=\"false\">(<\/mml:mo>\n      <mml:mi>q<\/mml:mi>\n      <mml:mo>\u2212<\/mml:mo>\n      <mml:mn>1<\/mml:mn>\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>m<\/mml:mi>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">f=f_{q}=(q-1)\/m<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>. We study the Jacobi sums <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper J Subscript a comma b Baseline equals minus sigma-summation Underscript k equals 2 Overscript q minus 1 Endscripts zeta Subscript m Superscript a ind Super Subscript s Superscript left-parenthesis k right-parenthesis plus b ind Super Subscript s Superscript left-parenthesis 1 minus k right-parenthesis\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:msub>\n        <mml:mi>J<\/mml:mi>\n        <mml:mrow class=\"MJX-TeXAtom-ORD\">\n          <mml:mi>a<\/mml:mi>\n          <mml:mo>,<\/mml:mo>\n          <mml:mi>b<\/mml:mi>\n        <\/mml:mrow>\n      <\/mml:msub>\n      <mml:mo>=<\/mml:mo>\n      <mml:mo>\u2212<\/mml:mo>\n      <mml:munderover>\n        <mml:mo>\u2211<\/mml:mo>\n        <mml:mrow class=\"MJX-TeXAtom-ORD\">\n          <mml:mi>k<\/mml:mi>\n          <mml:mo>=<\/mml:mo>\n          <mml:mn>2<\/mml:mn>\n        <\/mml:mrow>\n        <mml:mrow class=\"MJX-TeXAtom-ORD\">\n          <mml:mi>q<\/mml:mi>\n          <mml:mo>\u2212<\/mml:mo>\n          <mml:mn>1<\/mml:mn>\n        <\/mml:mrow>\n      <\/mml:munderover>\n      <mml:msubsup>\n        <mml:mi>\u03b6<\/mml:mi>\n        <mml:mi>m<\/mml:mi>\n        <mml:mrow class=\"MJX-TeXAtom-ORD\">\n          <mml:mspace width=\"thinmathspace\"\/>\n          <mml:mi>a<\/mml:mi>\n          <mml:mspace width=\"thinmathspace\"\/>\n          <mml:msub>\n            <mml:mtext>ind<\/mml:mtext>\n            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n              <mml:mi>s<\/mml:mi>\n            <\/mml:mrow>\n          <\/mml:msub>\n          <mml:mo stretchy=\"false\">(<\/mml:mo>\n          <mml:mi>k<\/mml:mi>\n          <mml:mo stretchy=\"false\">)<\/mml:mo>\n          <mml:mo>+<\/mml:mo>\n          <mml:mi>b<\/mml:mi>\n          <mml:mspace width=\"thinmathspace\"\/>\n          <mml:msub>\n            <mml:mtext>ind<\/mml:mtext>\n            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n              <mml:mi>s<\/mml:mi>\n            <\/mml:mrow>\n          <\/mml:msub>\n          <mml:mo stretchy=\"false\">(<\/mml:mo>\n          <mml:mn>1<\/mml:mn>\n          <mml:mo>\u2212<\/mml:mo>\n          <mml:mi>k<\/mml:mi>\n          <mml:mo stretchy=\"false\">)<\/mml:mo>\n        <\/mml:mrow>\n      <\/mml:msubsup>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">J_{a,b}=-\\sum _{k=2}^{q-1}\\zeta _m ^{\\, a\\, \\text {ind}_{s}(k)+b\\, \\text {ind}_{s}(1-k)}<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>, <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"0 less-than-or-equal-to a comma b less-than-or-equal-to m minus 1\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mn>0<\/mml:mn>\n      <mml:mo>\u2264<\/mml:mo>\n      <mml:mi>a<\/mml:mi>\n      <mml:mo>,<\/mml:mo>\n      <mml:mi>b<\/mml:mi>\n      <mml:mo>\u2264<\/mml:mo>\n      <mml:mi>m<\/mml:mi>\n      <mml:mo>\u2212<\/mml:mo>\n      <mml:mn>1<\/mml:mn>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">0\\leq a, b\\leq m-1<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>, where <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"ind Subscript s Baseline left-parenthesis k right-parenthesis\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:msub>\n        <mml:mtext>ind<\/mml:mtext>\n        <mml:mrow class=\"MJX-TeXAtom-ORD\">\n          <mml:mi>s<\/mml:mi>\n        <\/mml:mrow>\n      <\/mml:msub>\n      <mml:mo stretchy=\"false\">(<\/mml:mo>\n      <mml:mi>k<\/mml:mi>\n      <mml:mo stretchy=\"false\">)<\/mml:mo>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">\\text {ind}_{s}(k)<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> is the least nonnegative integer such that <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"s Superscript ind Super Subscript s Superscript left-parenthesis k right-parenthesis Baseline identical-to k\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:msup>\n        <mml:mi>s<\/mml:mi>\n        <mml:mrow class=\"MJX-TeXAtom-ORD\">\n          <mml:mspace width=\"thinmathspace\"\/>\n          <mml:msub>\n            <mml:mtext>ind<\/mml:mtext>\n            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n              <mml:mi>s<\/mml:mi>\n            <\/mml:mrow>\n          <\/mml:msub>\n          <mml:mo stretchy=\"false\">(<\/mml:mo>\n          <mml:mi>k<\/mml:mi>\n          <mml:mo stretchy=\"false\">)<\/mml:mo>\n        <\/mml:mrow>\n      <\/mml:msup>\n      <mml:mo>\u2261<\/mml:mo>\n      <mml:mi>k<\/mml:mi>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">s^{\\, \\text {ind}_{s}(k)}\\equiv k<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> mod <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>. We exhibit a set of properties that characterize these sums, some congruences they satisfy, and a MAPLE program to calculate them. Then we use those results to show how one can construct families <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper P Subscript q Baseline left-parenthesis x right-parenthesis\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:msub>\n        <mml:mi>P<\/mml:mi>\n        <mml:mrow class=\"MJX-TeXAtom-ORD\">\n          <mml:mi>q<\/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:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">P_{q}(x)<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>, <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"q element-of script upper P\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi>q<\/mml:mi>\n      <mml:mo>\u2208<\/mml:mo>\n      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n        <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">P<\/mml:mi>\n      <\/mml:mrow>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">q\\in \\mathcal {P}<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>, of irreducible polynomials of Gaussian periods, <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"eta Subscript i Baseline equals sigma-summation Underscript j equals 0 Overscript f minus 1 Endscripts zeta Subscript q Superscript s Super Superscript i plus m j\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:msub>\n        <mml:mi>\u03b7<\/mml:mi>\n        <mml:mrow class=\"MJX-TeXAtom-ORD\">\n          <mml:mi>i<\/mml:mi>\n        <\/mml:mrow>\n      <\/mml:msub>\n      <mml:mo>=<\/mml:mo>\n      <mml:munderover>\n        <mml:mo>\u2211<\/mml:mo>\n        <mml:mrow class=\"MJX-TeXAtom-ORD\">\n          <mml:mi>j<\/mml:mi>\n          <mml:mo>=<\/mml:mo>\n          <mml:mn>0<\/mml:mn>\n        <\/mml:mrow>\n        <mml:mrow class=\"MJX-TeXAtom-ORD\">\n          <mml:mi>f<\/mml:mi>\n          <mml:mo>\u2212<\/mml:mo>\n          <mml:mn>1<\/mml:mn>\n        <\/mml:mrow>\n      <\/mml:munderover>\n      <mml:msubsup>\n        <mml:mi>\u03b6<\/mml:mi>\n        <mml:mi>q<\/mml:mi>\n        <mml:mrow class=\"MJX-TeXAtom-ORD\">\n          <mml:msup>\n            <mml:mi>s<\/mml:mi>\n            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n              <mml:mi>i<\/mml:mi>\n              <mml:mo>+<\/mml:mo>\n              <mml:mi>m<\/mml:mi>\n              <mml:mi>j<\/mml:mi>\n            <\/mml:mrow>\n          <\/mml:msup>\n        <\/mml:mrow>\n      <\/mml:msubsup>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">\\eta _{i}=\\sum _{j=0}^{f-1}\\zeta _q^{s^{i+mj}}<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>, of degree <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>, where <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"script upper P\">\n  <mml:semantics>\n    <mml:mrow class=\"MJX-TeXAtom-ORD\">\n      <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">P<\/mml:mi>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">\\mathcal {P}<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> is a suitable set of primes <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"identical-to 1\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mo>\u2261<\/mml:mo>\n      <mml:mn>1<\/mml:mn>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">\\equiv 1<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> mod <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"2 m\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mn>2<\/mml:mn>\n      <mml:mi>m<\/mml:mi>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">2m<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>. We exhibit examples of such families for several small values of <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>, and give a MAPLE program to construct more of them.<\/p>","DOI":"10.1090\/s0025-5718-01-01312-6","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T22:13:53Z","timestamp":1027721633000},"page":"1617-1640","source":"Crossref","is-referenced-by-count":7,"title":["Jacobi sums and new families of irreducible polynomials of Gaussian periods"],"prefix":"10.1090","volume":"70","author":[{"given":"F.","family":"Thaine","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2001,5,11]]},"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","doi-asserted-by":"crossref","unstructured":"L.E. Dickson, Cyclotomy, higher congruences and Waring\u2019s problem, Amer. J. Math. 57 (1935), 391\u2013424.","DOI":"10.2307\/2371217"},{"key":"3","doi-asserted-by":"publisher","first-page":"99","DOI":"10.1093\/qmath\/os-10.1.99","article-title":"An algebraic property of Laplace\u2019s differential equation","volume":"10","author":"Taussky, Olga","year":"1939","journal-title":"Quart. J. Math. Oxford Ser.","ISSN":"https:\/\/id.crossref.org\/issn\/0033-5606","issn-type":"print"},{"key":"4","series-title":"Graduate Texts in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-0987-4","volume-title":"Cyclotomic fields I and II","volume":"121","author":"Lang, Serge","year":"1990","ISBN":"https:\/\/id.crossref.org\/isbn\/0387966714","edition":"2"},{"key":"5","doi-asserted-by":"publisher","first-page":"712","DOI":"10.2307\/1968951","article-title":"Rings with minimal condition for left ideals","volume":"40","author":"Hopkins, Charles","year":"1939","journal-title":"Ann. of Math. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0003-486X","issn-type":"print"},{"issue":"182","key":"6","doi-asserted-by":"publisher","first-page":"535","DOI":"10.2307\/2008622","article-title":"Connection between Gaussian periods and cyclic units","volume":"50","author":"Lehmer, Emma","year":"1988","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"182","key":"7","doi-asserted-by":"publisher","first-page":"535","DOI":"10.2307\/2008622","article-title":"Connection between Gaussian periods and cyclic units","volume":"50","author":"Lehmer, Emma","year":"1988","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"8","series-title":"Lectures in Advanced Mathematics, No. 2","volume-title":"Cyclotomy and difference sets","author":"Storer, Thomas","year":"1967"},{"key":"9","doi-asserted-by":"crossref","unstructured":"H.W. Lloyd Tanner, On the binomial equation \ud835\udc65^{\ud835\udc5d}-1=0: quinquisection, Proc. London Math. 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