zbMATH — the first resource for mathematics

On \(2\)-class field towers of some real quadratic number fields with \(2\)-class groups of rank \(3\). (English) Zbl 1302.11090
Summary: We construct an infinite family of real quadratic number fields with class group of 2-rank=3, 4-rank =1 and finite Hilbert 2-class field tower.

11R29 Class numbers, class groups, discriminants
11R32 Galois theory
11R37 Class field theory
Full Text: Euclid
[1] \beginbarticle \bauthor\binitsA. \bsnmAzizi and \bauthor\binitsA. \bsnmMouhib, \batitleSur le rang du 2-groupe de classes de \(\mathbf Q(\sqrt{m},\sqrt d)\) où \(m=2\) ou un premier \(p\equiv1\pmod4\), \bjtitleTrans. Amer. Math. Soc. \bvolume353 (\byear2001), no. \bissue7, page 2741-\blpage2752. \endbarticle \endbibitem · Zbl 0986.11073
[2] \beginbarticle \bauthor\binitsE. \bsnmBenjamin and \bauthor\binitsF. \bsnmLemmermeyer, \batitleReal quadratic fields with abelian \(2\)-class field tower, \bjtitleJ. Number Theory \bvolume73 (\byear1998), page 182-\blpage194. \endbarticle \endbibitem · Zbl 0919.11073
[3] \beginbarticle \bauthor\binitsE. \bsnmBenjamin and \bauthor\binitsC. \bsnmSnyder, \batitleReal quadratic number fields with \(2\)-class groupe of type \((2,2)\), \bjtitleMath. Scand. \bvolume76 (\byear1995), page 161-\blpage178. \endbarticle \endbibitem
[4] \beginbarticle \bauthor\binitsF. \bsnmGerth, \batitleQuadratic fields with infinite Hilbert 2-class field towers, \bjtitleActa Arith. \bvolume106 (\byear2003), page 151-\blpage158. \endbarticle \endbibitem · Zbl 1064.11073
[5] \beginbotherref \bauthor\binitsE. S. \bsnmGolod and \bauthor\binitsI. R. \bsnmShafarevich, On the class field tower , Izv. Akad. Nauk SSSR Ser. Math. 28 (1964), 261-272 (in Russian). English translation in AMS Transl. 48 (1965), 91-102. \endbotherref \endbibitem
[6] \beginbarticle \bauthor\binitsF. \bsnmHajir, \batitleOn a theorem of Koch, \bjtitlePacific J. Math. \bvolume176 (\byear1996), no. \bissue1, page 15-\blpage18. \endbarticle \endbibitem
[7] \beginbarticle \bauthor\binitsF. \bsnmHajir, \batitleCorrection to “On a theorem of Koch,” \bjtitlePacific J. Math. \bvolume196 (\byear2000), no. \bissue2, page 507-\blpage508. \endbarticle \endbibitem
[8] \beginbarticle \bauthor\binitsS. \bsnmKuroda, den Dirichletschen Körper, \bjtitleJ. Fac. Sci. Imp. Univ. Tokyo Sect. I \bvolume4 (\byear1943), page 383-\blpage406. \endbarticle \endbibitem
[9] \beginbarticle \bauthor\binitsF. \bsnmLemmermeyer, \batitleOn \(2\)-class field towers of some imaginary quadratic number fields, \bjtitleAbh. Math. Semin. Univ. Hambg. \bvolume67 (\byear1997), page 205-\blpage214. \endbarticle \endbibitem · Zbl 0919.11075
[10] \beginbotherref \oauthor\binitsF. \bsnmLemmermeyer, Higher descent on Pell conics. I. From Legendre to Selmer , available at \arxivurl
[11] \beginbarticle \bauthor\binitsA. \bsnmMouhib, \batitleOn the parity of the class number of real multiquadratic number fields, \bjtitleJ. Number. Theory \bvolume129 (\byear2009), page 1205-\blpage1211. \endbarticle \endbibitem · Zbl 1167.11039
[12] \beginbarticle \bauthor\binitsA. \bsnmMouhib, \batitleSur la borne inférieure du rang du 2-groupe de classes des corps multiquadratiques réels, \bjtitleCanad. Math. Bull. \bvolume54 (\byear2011), page 330-\blpage337. \endbarticle \endbibitem · Zbl 1248.11087
[13] \beginbarticle \bauthor\binitsL. , \batitleDie Anzahl der durch 4 teilbaren Invarienten der Klassengruppe eines beliebigen quadratischen Zahlkörpers, \bjtitleMath. Naturwiss. Anz. Ungar. Akad. d. Wiss. \bvolume49 (\byear1932), page 338-\blpage363. \endbarticle \endbibitem
[14] \beginbarticle \bauthor\binitsL. and \bauthor\binitsH. \bsnmReichardt, \batitleDie Anzahl der durch 4 teilbaren Invarienten der klassengruppe eines beliebigen quadratischen Zahlkörpers, \bjtitleJ. Reine Angew. Math. \bvolume170 (\byear1933), page 69-\blpage74. \endbarticle \endbibitem
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.