{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T21:52:38Z","timestamp":1776721958373,"version":"3.51.2"},"reference-count":15,"publisher":"American Mathematical Society (AMS)","issue":"216","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    Let\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L equals double-struck upper Q left-bracket alpha right-bracket\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>L<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi mathvariant=\"double-struck\">Q<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">[<\/mml:mo>\n                            <mml:mi>\n                              \u03b1\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">]<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">L={\\mathbb {Q}} [\\alpha ]<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    be an abelian number field of prime degree\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                    , and let\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"a\">\n                        <mml:semantics>\n                          <mml:mi>a<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">a<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    be a nonzero rational number. We describe an algorithm which takes as input\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"a\">\n                        <mml:semantics>\n                          <mml:mi>a<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">a<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and the minimal polynomial of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"alpha\">\n                        <mml:semantics>\n                          <mml:mi>\n                            \u03b1\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\alpha<\/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 Q\">\n                        <mml:semantics>\n                          <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"double-struck\">Q<\/mml:mi>\n                            <\/mml:mrow>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">{\\mathbb {Q}}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , and determines if\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"a\">\n                        <mml:semantics>\n                          <mml:mi>a<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">a<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is a norm of an element of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L\">\n                        <mml:semantics>\n                          <mml:mi>L<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">L<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . We show that, if we ignore the time needed to obtain a complete factorization of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"a\">\n                        <mml:semantics>\n                          <mml:mi>a<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">a<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and a complete factorization of the discriminant of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"alpha\">\n                        <mml:semantics>\n                          <mml:mi>\n                            \u03b1\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\alpha<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , then the algorithm runs in time polynomial in the size of the input. As an application, we give an algorithm to test if a cyclic algebra\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper A equals left-parenthesis upper E comma sigma comma a right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>A<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>E<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>\n                              \u03c3\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\">A=( E, \\sigma , a )<\/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 Q\">\n                        <mml:semantics>\n                          <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"double-struck\">Q<\/mml:mi>\n                            <\/mml:mrow>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">{\\mathbb {Q}}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is a division algebra.\n                  <\/p>","DOI":"10.1090\/s0025-5718-96-00760-0","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:44Z","timestamp":1027707284000},"page":"1663-1674","source":"Crossref","is-referenced-by-count":5,"title":["Solvability of norm equations over cyclic number fields of prime degree"],"prefix":"10.1090","volume":"65","author":[{"given":"Vincenzo","family":"Acciaro","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[1996]]},"reference":[{"key":"1","series-title":"American Mathematical Society Colloquium Publications, Vol. XXIV","volume-title":"Structure of algebras","author":"Albert, A. 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