{"title":"Capitulation of the 2-ideal classes of type (2, 2, 2) of some quartic cyclic number fields","authors":"A. Azizi, I. Jerrari, A. Zekhnini, M. Talbi","doi":"10.1515/jmc-2017-0037","DOIUrl":null,"url":null,"abstract":"Abstract Let p ≡ 3 ( mod 4 ) {p\\equiv 3\\pmod{4}} and l ≡ 5 ( mod 8 ) {l\\equiv 5\\pmod{8}} be different primes such that p l = 1 {\\frac{p}{l}=1} and 2 p = p l 4 {\\frac{2}{p}=\\frac{p}{l}_{4}} . Put k = ℚ ( l ) {k=\\mathbb{Q}(\\sqrt{l})} , and denote by ϵ its fundamental unit. Set K = k ( - 2 p ϵ l ) {K=k(\\sqrt{-2p\\epsilon\\sqrt{l}})} , and let K 2 ( 1 ) {K_{2}^{(1)}} be its Hilbert 2-class field, and let K 2 ( 2 ) {K_{2}^{(2)}} be its second Hilbert 2-class field. The field K is a cyclic quartic number field, and its 2-class group is of type ( 2 , 2 , 2 ) {(2,2,2)} . Our goal is to prove that the length of the 2-class field tower of K is 2, to determine the structure of the 2-group G = Gal ( K 2 ( 2 ) / K ) {G=\\operatorname{Gal}(K_{2}^{(2)}/K)} , and thus to study the capitulation of the 2-ideal classes of K in all its unramified abelian extensions within K 2 ( 1 ) {K_{2}^{(1)}} . Additionally, these extensions are constructed, and their abelian-type invariants are given.","PeriodicalId":43866,"journal":{"name":"Journal of Mathematical Cryptology","volume":null,"pages":null},"PeriodicalIF":0.5000,"publicationDate":"2019-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/jmc-2017-0037","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Mathematical Cryptology","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1515/jmc-2017-0037","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
引用次数: 0
Abstract
Abstract Let p ≡ 3 ( mod 4 ) {p\equiv 3\pmod{4}} and l ≡ 5 ( mod 8 ) {l\equiv 5\pmod{8}} be different primes such that p l = 1 {\frac{p}{l}=1} and 2 p = p l 4 {\frac{2}{p}=\frac{p}{l}_{4}} . Put k = ℚ ( l ) {k=\mathbb{Q}(\sqrt{l})} , and denote by ϵ its fundamental unit. Set K = k ( - 2 p ϵ l ) {K=k(\sqrt{-2p\epsilon\sqrt{l}})} , and let K 2 ( 1 ) {K_{2}^{(1)}} be its Hilbert 2-class field, and let K 2 ( 2 ) {K_{2}^{(2)}} be its second Hilbert 2-class field. The field K is a cyclic quartic number field, and its 2-class group is of type ( 2 , 2 , 2 ) {(2,2,2)} . Our goal is to prove that the length of the 2-class field tower of K is 2, to determine the structure of the 2-group G = Gal ( K 2 ( 2 ) / K ) {G=\operatorname{Gal}(K_{2}^{(2)}/K)} , and thus to study the capitulation of the 2-ideal classes of K in all its unramified abelian extensions within K 2 ( 1 ) {K_{2}^{(1)}} . Additionally, these extensions are constructed, and their abelian-type invariants are given.