{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T09:37:39Z","timestamp":1776764259047,"version":"3.51.2"},"reference-count":11,"publisher":"American Mathematical Society (AMS)","issue":"214","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=\"k\">\n                        <mml:semantics>\n                          <mml:mi>k<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">k<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    be a real quadratic field and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p\">\n                        <mml:semantics>\n                          <mml:mi>p<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">p<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    an odd prime number which splits in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k\">\n                        <mml:semantics>\n                          <mml:mi>k<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">k<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . In a previous work, the author gave a sufficient condition for the Iwasawa invariant\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"lamda Subscript p Baseline left-parenthesis k right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>\n                                \u03bb\n                                \n                              <\/mml:mi>\n                              <mml:mi>p<\/mml:mi>\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\">\\lambda _p(k)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    of the cyclotomic\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"double-struck upper Z Subscript p\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"double-struck\">Z<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mi>p<\/mml:mi>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathbb {Z}_p<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -extension of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k\">\n                        <mml:semantics>\n                          <mml:mi>k<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">k<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    to be zero. The purpose of this paper is to study the case\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p equals 3\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>p<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>3<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">p=3<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    of this result and give new examples of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k\">\n                        <mml:semantics>\n                          <mml:mi>k<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">k<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    with\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"lamda 3 left-parenthesis k right-parenthesis equals 0\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>\n                                \u03bb\n                                \n                              <\/mml:mi>\n                              <mml:mn>3<\/mml:mn>\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:mn>0<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\lambda _3(k)=0<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , by using information on the initial layer of the cyclotomic\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"double-struck upper Z 3\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"double-struck\">Z<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mn>3<\/mml:mn>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathbb {Z}_3<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -extension of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k\">\n                        <mml:semantics>\n                          <mml:mi>k<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">k<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    .\n                  <\/p>","DOI":"10.1090\/s0025-5718-96-00721-1","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:44Z","timestamp":1027707284000},"page":"779-784","source":"Crossref","is-referenced-by-count":4,"title":["Computation of \u2124\u2083-invariants of real quadratic fields"],"prefix":"10.1090","volume":"65","author":[{"given":"Hisao","family":"Taya","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[1996]]},"reference":[{"issue":"2","key":"1","doi-asserted-by":"publisher","first-page":"377","DOI":"10.2307\/1971116","article-title":"The Iwasawa invariant \ud835\udf07_{\ud835\udc5d} vanishes for abelian number fields","volume":"109","author":"Ferrero, Bruce","year":"1979","journal-title":"Ann. of Math. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0003-486X","issn-type":"print"},{"issue":"7","key":"2","first-page":"260","article-title":"Iwasawa\u2019s \ud835\udf06-invariants of certain real quadratic fields","volume":"65","author":"Fukuda, Takashi","year":"1989","journal-title":"Proc. Japan Acad. Ser. A Math. Sci.","ISSN":"https:\/\/id.crossref.org\/issn\/0386-2194","issn-type":"print"},{"issue":"1","key":"3","first-page":"27","article-title":"Computation of unit group for \ud835\udc4d\u2083-extensions of real quadratic fields","volume":"13","author":"Fukuda, Takashi","year":"1992","journal-title":"Bull. Yamagata Univ. Natur. Sci.","ISSN":"https:\/\/id.crossref.org\/issn\/0513-4692","issn-type":"print"},{"key":"4","unstructured":"\\bysame, Iwasawa \ud835\udf06-invariants of imaginary quadratic fields, J. of the College of Industrial Technology, Nihon Univ. 27 (1994), 35\u201388."},{"issue":"1","key":"5","doi-asserted-by":"publisher","first-page":"95","DOI":"10.2969\/jmsj\/03810095","article-title":"On \ud835\udc4d_{\ud835\udc5d}-extensions of real quadratic fields","volume":"38","author":"Fukuda, Takashi","year":"1986","journal-title":"J. Math. Soc. Japan","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5645","issn-type":"print"},{"issue":"8","key":"6","first-page":"318","article-title":"A remark on the \ud835\udf06-invariant of real quadratic fields","volume":"62","author":"Fukuda, Takashi","year":"1986","journal-title":"Proc. Japan Acad. Ser. A Math. Sci.","ISSN":"https:\/\/id.crossref.org\/issn\/0386-2194","issn-type":"print"},{"issue":"1","key":"7","doi-asserted-by":"publisher","first-page":"263","DOI":"10.2307\/2373625","article-title":"On the Iwasawa invariants of totally real number fields","volume":"98","author":"Greenberg, Ralph","year":"1976","journal-title":"Amer. J. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9327","issn-type":"print"},{"key":"8","doi-asserted-by":"publisher","first-page":"246","DOI":"10.2307\/1970784","article-title":"On \ud835\udc4d_{\ud835\udc59}-extensions of algebraic number fields","volume":"98","author":"Iwasawa, Kenkichi","year":"1973","journal-title":"Ann. of Math. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0003-486X","issn-type":"print"},{"key":"9","series-title":"Lecture Notes in Mathematics","isbn-type":"print","doi-asserted-by":"crossref","DOI":"10.1007\/BFb0088938","volume-title":"The determination of units in real cyclic sextic fields","volume":"797","author":"M\u00e4ki, Sirpa","year":"1980","ISBN":"https:\/\/id.crossref.org\/isbn\/3540099840"},{"issue":"1","key":"10","doi-asserted-by":"publisher","first-page":"121","DOI":"10.3836\/tjm\/1270128985","article-title":"On the Iwasawa \ud835\udf06-invariants of real quadratic fields","volume":"16","author":"Taya, Hisao","year":"1993","journal-title":"Tokyo J. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0387-3870","issn-type":"print"},{"key":"11","series-title":"Graduate Texts in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4684-0133-2","volume-title":"Introduction to cyclotomic fields","volume":"83","author":"Washington, Lawrence C.","year":"1982","ISBN":"https:\/\/id.crossref.org\/isbn\/0387906223"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/1996-65-214\/S0025-5718-96-00721-1\/S0025-5718-96-00721-1.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/1996-65-214\/S0025-5718-96-00721-1\/S0025-5718-96-00721-1.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T21:07:27Z","timestamp":1776719247000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/1996-65-214\/S0025-5718-96-00721-1\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1996]]},"references-count":11,"journal-issue":{"issue":"214","published-print":{"date-parts":[[1996,4]]}},"alternative-id":["S0025-5718-96-00721-1"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-96-00721-1","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":[[1996]]}}}