{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,1]],"date-time":"2026-08-01T00:05:19Z","timestamp":1785542719424,"version":"3.56.0"},"reference-count":21,"publisher":"American Mathematical Society (AMS)","issue":"276","license":[{"start":{"date-parts":[[2012,4,11]],"date-time":"2012-04-11T00:00:00Z","timestamp":1334102400000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n Let\n \n \n \n E<\/mml:mi>\n E<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n be a number field and\n \n \n \n G<\/mml:mi>\n G<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n a finite group. Let\n \n \n \n \n A<\/mml:mi>\n <\/mml:mrow>\n \\mathcal {A}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n be any\n \n \n \n \n \n O<\/mml:mi>\n <\/mml:mrow>\n \n E<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n \\mathcal {O}_{E}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -order of full rank in the group algebra\n \n \n \n \n E<\/mml:mi>\n [<\/mml:mo>\n G<\/mml:mi>\n ]<\/mml:mo>\n <\/mml:mrow>\n E[G]<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\n \n \n \n X<\/mml:mi>\n X<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n a (left)\n \n \n \n \n A<\/mml:mi>\n <\/mml:mrow>\n \\mathcal {A}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -lattice. In a previous article, we gave a necessary and sufficient condition for\n \n \n \n X<\/mml:mi>\n X<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n to be free of given rank\n \n \n \n d<\/mml:mi>\n d<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n over\n \n \n \n \n A<\/mml:mi>\n <\/mml:mrow>\n \\mathcal {A}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . In the case that (i) the Wedderburn decomposition\n \n \n \n \n E<\/mml:mi>\n [<\/mml:mo>\n G<\/mml:mi>\n ]<\/mml:mo>\n \n \u2245\n \n <\/mml:mo>\n \n \n \u2a01\n \n <\/mml:mo>\n \n \n \u03c7\n \n <\/mml:mi>\n <\/mml:mrow>\n <\/mml:munder>\n \n M<\/mml:mi>\n \n \n \u03c7\n \n <\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:mrow>\n E[G] \\cong \\bigoplus _{\\chi } M_{\\chi }<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is explicitly computable and (ii) each\n \n \n \n \n M<\/mml:mi>\n \n \n \u03c7\n \n <\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n M_{\\chi }<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is in fact a matrix ring over a field, this led to an algorithm that either gives elements\n \n \n \n \n \n \n \u03b1\n \n <\/mml:mi>\n \n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n ,<\/mml:mo>\n \n \u2026\n \n <\/mml:mo>\n ,<\/mml:mo>\n \n \n \u03b1\n \n <\/mml:mi>\n \n d<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n \n \u2208\n \n <\/mml:mo>\n X<\/mml:mi>\n <\/mml:mrow>\n \\alpha _{1}, \\ldots , \\alpha _{d} \\in X<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n such that\n \n \n \n \n X<\/mml:mi>\n =<\/mml:mo>\n \n A<\/mml:mi>\n <\/mml:mrow>\n \n \n \u03b1\n \n <\/mml:mi>\n \n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n \n \u2295\n \n <\/mml:mo>\n \n \u22ef\n \n <\/mml:mo>\n \n \u2295\n \n <\/mml:mo>\n \n A<\/mml:mi>\n <\/mml:mrow>\n \n \n \u03b1\n \n <\/mml:mi>\n \n d<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:mrow>\n X=\\mathcal {A}\\alpha _{1} \\oplus \\cdots \\oplus \\mathcal {A}\\alpha _{d}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n or determines that no such elements exist. In the present article, we generalise the algorithm by weakening condition (ii) considerably.\n <\/p>","DOI":"10.1090\/s0025-5718-2011-02488-9","type":"journal-article","created":{"date-parts":[[2011,4,11]],"date-time":"2011-04-11T14:27:29Z","timestamp":1302532049000},"page":"2411-2434","source":"Crossref","is-referenced-by-count":6,"title":["Computing generators of free modules over orders in group algebras II"],"prefix":"10.1090","volume":"80","author":[{"given":"Werner","family":"Bley","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Henri","family":"Johnston","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2011,4,11]]},"reference":[{"key":"1","isbn-type":"print","doi-asserted-by":"publisher","first-page":"72","DOI":"10.1007\/11792086_6","article-title":"Computation of locally free class groups","author":"Bley, Werner","year":"2006","ISBN":"https:\/\/id.crossref.org\/isbn\/9783540360759"},{"issue":"3-4","key":"2","doi-asserted-by":"publisher","first-page":"235","DOI":"10.1006\/jsco.1996.0125","article-title":"The Magma algebra system. 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