{"title":"数域上的可容许群","authors":"Deependra Singh","doi":"arxiv-2409.02333","DOIUrl":null,"url":null,"abstract":"Given a field K, one may ask which finite groups are Galois groups of field\nextensions L/K such that L is a maximal subfield of a division algebra with\ncenter K. This connection between inverse Galois theory and division algebras\nwas first explored by Schacher in the 1960s. In this manuscript we consider\nthis problem when K is a number field. For the case when L/K is assumed to be\ntamely ramified, we give a complete classification of number fields for which\nevery solvable Sylow-metacyclic group is admissible, extending J. Sonn's result\nover the field of rational numbers. For the case when L/K is allowed to be\nwildly ramified, we give a characterization of admissible groups over several\nclasses of number fields, and partial results in other cases.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"35 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Admissible groups over number fields\",\"authors\":\"Deependra Singh\",\"doi\":\"arxiv-2409.02333\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Given a field K, one may ask which finite groups are Galois groups of field\\nextensions L/K such that L is a maximal subfield of a division algebra with\\ncenter K. This connection between inverse Galois theory and division algebras\\nwas first explored by Schacher in the 1960s. In this manuscript we consider\\nthis problem when K is a number field. For the case when L/K is assumed to be\\ntamely ramified, we give a complete classification of number fields for which\\nevery solvable Sylow-metacyclic group is admissible, extending J. Sonn's result\\nover the field of rational numbers. For the case when L/K is allowed to be\\nwildly ramified, we give a characterization of admissible groups over several\\nclasses of number fields, and partial results in other cases.\",\"PeriodicalId\":501136,\"journal\":{\"name\":\"arXiv - MATH - Rings and Algebras\",\"volume\":\"35 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-03\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Rings and Algebras\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2409.02333\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Rings and Algebras","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.02333","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
摘要
给定一个域 K,我们可能会问,哪些有限群是域扩展 L/K 的伽罗瓦群,从而使 L 成为以 K 为中心的除法代数的最大子域?在本手稿中,我们考虑的是 K 为数域时的问题。对于假定 L/K 完全夯化的情况,我们给出了一个完整的数域分类,对于这些数域,每个可解的 Sylow-metacyclic 群都是可容许的,从而扩展了 J. Sonn 在有理数域上的结果。对于允许 L/K 任意横切的情况,我们给出了几类数域上可容许群的特征,并给出了其他情况下的部分结果。
Given a field K, one may ask which finite groups are Galois groups of field
extensions L/K such that L is a maximal subfield of a division algebra with
center K. This connection between inverse Galois theory and division algebras
was first explored by Schacher in the 1960s. In this manuscript we consider
this problem when K is a number field. For the case when L/K is assumed to be
tamely ramified, we give a complete classification of number fields for which
every solvable Sylow-metacyclic group is admissible, extending J. Sonn's result
over the field of rational numbers. For the case when L/K is allowed to be
wildly ramified, we give a characterization of admissible groups over several
classes of number fields, and partial results in other cases.