{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T23:45:33Z","timestamp":1776728733293,"version":"3.51.2"},"reference-count":16,"publisher":"American Mathematical Society (AMS)","issue":"242","license":[{"start":{"date-parts":[[2003,6,4]],"date-time":"2003-06-04T00:00:00Z","timestamp":1054684800000},"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 We say a tame Galois field extension\n \n \n \n \n L<\/mml:mi>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n K<\/mml:mi>\n <\/mml:mrow>\n L\/K<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n with Galois group\n \n \n \n G<\/mml:mi>\n G<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n has trivial Galois module structure if the rings of integers have the property that\n \n \n \n \n \n O<\/mml:mi>\n <\/mml:mrow>\n \n L<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n \\mathcal {O}_{L}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is a free\n \n \n \n \n \n \n O<\/mml:mi>\n <\/mml:mrow>\n \n K<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n [<\/mml:mo>\n G<\/mml:mi>\n ]<\/mml:mo>\n <\/mml:mrow>\n \\mathcal {O}_{K}[G]<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -module. The work of Greither, Replogle, Rubin, and Srivastav shows that for each algebraic number field other than the rational numbers there will exist infinitely many primes\n \n \n \n l<\/mml:mi>\n l<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n so that for each there is a tame Galois field extension of degree\n \n \n \n l<\/mml:mi>\n l<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n so that\n \n \n \n \n L<\/mml:mi>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n K<\/mml:mi>\n <\/mml:mrow>\n L\/K<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n has nontrivial Galois module structure. However, the proof does not directly yield specific primes\n \n \n \n l<\/mml:mi>\n l<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n for a given algebraic number field\n \n \n \n \n K<\/mml:mi>\n .<\/mml:mo>\n <\/mml:mrow>\n K.<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n For\n \n \n \n K<\/mml:mi>\n K<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n any cyclotomic field we find an explicit\n \n \n \n l<\/mml:mi>\n l<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n so that there is a tame degree\n \n \n \n l<\/mml:mi>\n l<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n extension\n \n \n \n \n L<\/mml:mi>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n K<\/mml:mi>\n <\/mml:mrow>\n L\/K<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n with nontrivial Galois module structure.\n <\/p>","DOI":"10.1090\/s0025-5718-02-01457-6","type":"journal-article","created":{"date-parts":[[2003,2,10]],"date-time":"2003-02-10T11:10:52Z","timestamp":1044875452000},"page":"891-899","source":"Crossref","is-referenced-by-count":4,"title":["Nontrivial Galois module structure of cyclotomic fields"],"prefix":"10.1090","volume":"72","author":[{"given":"Marc","family":"Conrad","sequence":"first","affiliation":[]},{"given":"Daniel","family":"Replogle","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2002,6,4]]},"reference":[{"key":"1","doi-asserted-by":"publisher","first-page":"205","DOI":"10.1515\/crll.1993.434.205","article-title":"Relative Galois module structure of rings of integers of cyclotomic fields","volume":"434","author":"Chan, Shih-Ping","year":"1993","journal-title":"J. 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