{"title":"实数循环四次域的Hilbert属域的构造","authors":"M. M. Chems-Eddin, Moulay Ahmed Hajjami, M. Taous","doi":"10.7169/facm/2014","DOIUrl":null,"url":null,"abstract":"Let k be a number field and let H(k) denote the Hilbert class field of k, that is the maximal abelian unramified extension of k. It is known by class field theory that the Galois group of the extension H(k)/k, i.e., G := Gal(H(k)/k), is isomorphic to Cl(k), the class group of k (cf. [13, p. 228]). The Hilbert genus field of k, denoted by E(k), is the invariant field of G. Thus, by Galois theory, we have: Cl(k)/Cl(k) ≃ G/G ≃ Gal(E(k)/k),","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":null,"pages":null},"PeriodicalIF":0.5000,"publicationDate":"2021-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The construction of the Hilbert genus fields of real cyclic quartic fields\",\"authors\":\"M. M. Chems-Eddin, Moulay Ahmed Hajjami, M. Taous\",\"doi\":\"10.7169/facm/2014\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let k be a number field and let H(k) denote the Hilbert class field of k, that is the maximal abelian unramified extension of k. It is known by class field theory that the Galois group of the extension H(k)/k, i.e., G := Gal(H(k)/k), is isomorphic to Cl(k), the class group of k (cf. [13, p. 228]). The Hilbert genus field of k, denoted by E(k), is the invariant field of G. Thus, by Galois theory, we have: Cl(k)/Cl(k) ≃ G/G ≃ Gal(E(k)/k),\",\"PeriodicalId\":44655,\"journal\":{\"name\":\"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.5000,\"publicationDate\":\"2021-05-24\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.7169/facm/2014\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.7169/facm/2014","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
设k为数域，H(k)表示k的Hilbert类域，即k的最大阿贝尔无分支扩展。由类场论可知，扩展H(k)/k的伽罗瓦群，即G:= Gal(H(k)/k)，与k的类群Cl(k)同构(cf. [13, p. 228])。k的Hilbert格场，用E(k)表示，是G的不变场。因此，根据伽罗瓦理论，我们得到:Cl(k)/Cl(k)≃G/G≃Gal(E(k)/k);
The construction of the Hilbert genus fields of real cyclic quartic fields
Let k be a number field and let H(k) denote the Hilbert class field of k, that is the maximal abelian unramified extension of k. It is known by class field theory that the Galois group of the extension H(k)/k, i.e., G := Gal(H(k)/k), is isomorphic to Cl(k), the class group of k (cf. [13, p. 228]). The Hilbert genus field of k, denoted by E(k), is the invariant field of G. Thus, by Galois theory, we have: Cl(k)/Cl(k) ≃ G/G ≃ Gal(E(k)/k),
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