{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T07:48:57Z","timestamp":1776844137621,"version":"3.51.2"},"reference-count":23,"publisher":"American Mathematical Society (AMS)","issue":"242","license":[{"start":{"date-parts":[[2003,2,15]],"date-time":"2003-02-15T00:00:00Z","timestamp":1045267200000},"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":"

\n The class numbers\n \n \n \n \n h<\/mml:mi>\n \n l<\/mml:mi>\n <\/mml:mrow>\n \n +<\/mml:mo>\n <\/mml:mrow>\n <\/mml:msubsup>\n h_{l}^{+}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n of the real cyclotomic fields\n \n \n \n \n \n Q<\/mml:mi>\n <\/mml:mrow>\n (<\/mml:mo>\n \n \n \u03b6\n \n <\/mml:mi>\n \n l<\/mml:mi>\n <\/mml:mrow>\n \n <\/mml:msubsup>\n +<\/mml:mo>\n \n \n \u03b6\n \n <\/mml:mi>\n \n l<\/mml:mi>\n <\/mml:mrow>\n \n \n \u2212\n \n <\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msubsup>\n )<\/mml:mo>\n <\/mml:mrow>\n \\mathbf {Q}(\\zeta _{l}^{}+\\zeta _{l}^{-1})<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n are notoriously hard to compute. Indeed, the number\n \n \n \n \n h<\/mml:mi>\n \n l<\/mml:mi>\n <\/mml:mrow>\n \n +<\/mml:mo>\n <\/mml:mrow>\n <\/mml:msubsup>\n h_{l}^{+}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is not known for a single prime\n \n \n \n \n l<\/mml:mi>\n \n \u2265\n \n <\/mml:mo>\n 71<\/mml:mn>\n <\/mml:mrow>\n l\\ge 71<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . In this paper we present a table of the orders of certain subgroups of the class groups of the real cyclotomic fields\n \n \n \n \n \n Q<\/mml:mi>\n <\/mml:mrow>\n (<\/mml:mo>\n \n \n \u03b6\n \n <\/mml:mi>\n \n l<\/mml:mi>\n <\/mml:mrow>\n \n <\/mml:msubsup>\n +<\/mml:mo>\n \n \n \u03b6\n \n <\/mml:mi>\n \n l<\/mml:mi>\n <\/mml:mrow>\n \n \n \u2212\n \n <\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msubsup>\n )<\/mml:mo>\n <\/mml:mrow>\n \\mathbf {Q}(\\zeta _{l}^{}+\\zeta _{l}^{-1})<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n for the primes\n \n \n \n \n l<\/mml:mi>\n ><\/mml:mo>\n 10<\/mml:mn>\n ,<\/mml:mo>\n 000<\/mml:mn>\n <\/mml:mrow>\n l>10,000<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . It is quite likely that these subgroups are in fact\n equal<\/italic>\n to the class groups themselves, but there is at present no hope of proving this rigorously. In the last section of the paper we argue \u2014on the basis of the Cohen-Lenstra heuristics\u2014 that the probability that our table is actually a table of class numbers\n \n \n \n \n h<\/mml:mi>\n \n l<\/mml:mi>\n <\/mml:mrow>\n \n +<\/mml:mo>\n <\/mml:mrow>\n <\/mml:msubsup>\n h_{l}^{+}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , is at least\n \n \n \n 98<\/mml:mn>\n 98%<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n .\n <\/p>","DOI":"10.1090\/s0025-5718-02-01432-1","type":"journal-article","created":{"date-parts":[[2003,2,10]],"date-time":"2003-02-10T11:10:52Z","timestamp":1044875452000},"page":"913-937","source":"Crossref","is-referenced-by-count":29,"title":["Class numbers of real cyclotomic fields of prime conductor"],"prefix":"10.1090","volume":"72","author":[{"given":"Ren\u00e9","family":"Schoof","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2002,2,15]]},"reference":[{"key":"1","unstructured":"[1] Batut, C., Belabas, K., Bernardi, D., Cohen, H., Olivier, M.: User\u2019s guide to PARI-GP, (version 2.0.16), Lab. 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