{"title":"具有2-幂次Frobenius-Galois群的Galois推广中的算术等价域","authors":"Masanari Kida","doi":"10.4153/S0008439522000388","DOIUrl":null,"url":null,"abstract":"Abstract Let \n$F_{2^n}$\n be the Frobenius group of degree \n$2^n$\n and of order \n$2^n ( 2^n-1)$\n with \n$n \\ge 4$\n . We show that if \n$K/\\mathbb {Q} $\n is a Galois extension whose Galois group is isomorphic to \n$F_{2^n}$\n , then there are \n$\\dfrac {2^{n-1} +(-1)^n }{3}$\n intermediate fields of \n$K/\\mathbb {Q} $\n of degree \n$4 (2^n-1)$\n such that they are not conjugate over \n$\\mathbb {Q}$\n but arithmetically equivalent over \n$\\mathbb {Q}$\n . We also give an explicit method to construct these arithmetically equivalent fields.","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Arithmetically equivalent fields in a Galois extension with Frobenius Galois group of 2-power degree\",\"authors\":\"Masanari Kida\",\"doi\":\"10.4153/S0008439522000388\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Abstract Let \\n$F_{2^n}$\\n be the Frobenius group of degree \\n$2^n$\\n and of order \\n$2^n ( 2^n-1)$\\n with \\n$n \\\\ge 4$\\n . We show that if \\n$K/\\\\mathbb {Q} $\\n is a Galois extension whose Galois group is isomorphic to \\n$F_{2^n}$\\n , then there are \\n$\\\\dfrac {2^{n-1} +(-1)^n }{3}$\\n intermediate fields of \\n$K/\\\\mathbb {Q} $\\n of degree \\n$4 (2^n-1)$\\n such that they are not conjugate over \\n$\\\\mathbb {Q}$\\n but arithmetically equivalent over \\n$\\\\mathbb {Q}$\\n . We also give an explicit method to construct these arithmetically equivalent fields.\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2022-06-13\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.4153/S0008439522000388\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4153/S0008439522000388","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Arithmetically equivalent fields in a Galois extension with Frobenius Galois group of 2-power degree
Abstract Let
$F_{2^n}$
be the Frobenius group of degree
$2^n$
and of order
$2^n ( 2^n-1)$
with
$n \ge 4$
. We show that if
$K/\mathbb {Q} $
is a Galois extension whose Galois group is isomorphic to
$F_{2^n}$
, then there are
$\dfrac {2^{n-1} +(-1)^n }{3}$
intermediate fields of
$K/\mathbb {Q} $
of degree
$4 (2^n-1)$
such that they are not conjugate over
$\mathbb {Q}$
but arithmetically equivalent over
$\mathbb {Q}$
. We also give an explicit method to construct these arithmetically equivalent fields.