{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T10:53:46Z","timestamp":1776768826266,"version":"3.51.2"},"reference-count":24,"publisher":"American Mathematical Society (AMS)","issue":"223","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n We show that for any prime number\n \n \n \n \n l<\/mml:mi>\n ><\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n l>2<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n the minus class group of the field of the\n \n \n \n l<\/mml:mi>\n l<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -th roots of unity\n \n \n \n \n \n \n \n Q<\/mml:mi>\n <\/mml:mrow>\n p<\/mml:mi>\n <\/mml:msub>\n \n \u00af\n \n <\/mml:mo>\n <\/mml:mover>\n (<\/mml:mo>\n \n \n \u03b6\n \n <\/mml:mi>\n l<\/mml:mi>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n \\overline {\\mathbf {Q}_p} (\\zeta _l)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n admits a finite free resolution of length\u00a01 as a module over the ring\n \n \n \n \n \n \n \n Z<\/mml:mi>\n <\/mml:mrow>\n \n ^\n \n <\/mml:mo>\n <\/mml:mover>\n <\/mml:mrow>\n [<\/mml:mo>\n G<\/mml:mi>\n ]<\/mml:mo>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n (<\/mml:mo>\n 1<\/mml:mn>\n +<\/mml:mo>\n \n \u03b9\n \n <\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \\widehat {\\mathbf {Z}} [G]\/(1+\\iota )<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . Here\n \n \n \n \n \u03b9\n \n <\/mml:mi>\n \\iota<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n denotes complex conjugation in\n \n \n \n \n G<\/mml:mi>\n =<\/mml:mo>\n \n \n G<\/mml:mi>\n a<\/mml:mi>\n l<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n (<\/mml:mo>\n \n \n \n Q<\/mml:mi>\n <\/mml:mrow>\n p<\/mml:mi>\n <\/mml:msub>\n \n \u00af\n \n <\/mml:mo>\n <\/mml:mover>\n (<\/mml:mo>\n \n \n \u03b6\n \n <\/mml:mi>\n l<\/mml:mi>\n <\/mml:msub>\n )<\/mml:mo>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n \n \n \n Q<\/mml:mi>\n <\/mml:mrow>\n p<\/mml:mi>\n <\/mml:msub>\n \n \u00af\n \n <\/mml:mo>\n <\/mml:mover>\n )<\/mml:mo>\n \n \u2245\n \n <\/mml:mo>\n (<\/mml:mo>\n \n Z<\/mml:mi>\n <\/mml:mrow>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n l<\/mml:mi>\n \n Z<\/mml:mi>\n <\/mml:mrow>\n \n )<\/mml:mo>\n \n \u2217\n \n <\/mml:mo>\n <\/mml:msup>\n <\/mml:mrow>\n G={{Gal}}( \\overline {\\mathbf {Q}_p} (\\zeta _l)\/\\overline {\\mathbf {Q}_p} )\\cong (\\mathbf {Z} \/l\\mathbf {Z} )^*<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . Moreover, for the primes\n \n \n \n \n l<\/mml:mi>\n \n \u2264\n \n <\/mml:mo>\n 509<\/mml:mn>\n <\/mml:mrow>\n l\\le 509<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n we show that the minus class group is cyclic as a module over this ring. For these primes we also determine the structure of the minus class group.\n <\/p>","DOI":"10.1090\/s0025-5718-98-00939-9","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:13:45Z","timestamp":1027707225000},"page":"1225-1245","source":"Crossref","is-referenced-by-count":20,"title":["Minus class groups of the fields of the \ud835\udc59-th roots of unity"],"prefix":"10.1090","volume":"67","author":[{"given":"Ren\u00e9","family":"Schoof","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[1998]]},"reference":[{"key":"1","unstructured":"[1] Bourbaki, N.: \u00c9l\u00e9ments de Math\u00e9matique, Alg\u00e8bre, Hermann, Paris 1970."},{"key":"2","volume-title":"Algebraic number theory","year":"1967"},{"key":"3","unstructured":"[3] Cornacchia, P.: Anderson\u2019s module and ideal class groups of abelian fields, J. 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No. 717, 69--106","article-title":"Travaux de Kolyvagin et Rubin","author":"Perrin-Riou, Bernadette","year":"1990","journal-title":"Ast\\'{e}risque","ISSN":"https:\/\/id.crossref.org\/issn\/0303-1179","issn-type":"print"},{"key":"18","isbn-type":"print","doi-asserted-by":"publisher","first-page":"309","DOI":"10.1007\/978-1-4612-0457-2_14","article-title":"Kolyvagin\u2019s system of Gauss sums","author":"Rubin, Karl","year":"1991","ISBN":"https:\/\/id.crossref.org\/isbn\/0817635130"},{"issue":"1-2","key":"19","doi-asserted-by":"publisher","first-page":"125","DOI":"10.1016\/0022-4049(88)90016-3","article-title":"Cohomology of class groups of cyclotomic fields: an application to Morse-Smale diffeomorphisms","volume":"53","author":"Schoof, Ren\u00e9","year":"1988","journal-title":"J. Pure Appl. 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