×

zbMATH — the first resource for mathematics

Maximal unramified extensions of imaginary quadratic number fields of small conductors. II. (English) Zbl 1013.11076
This paper is an extension of a previous paper by the author [Part I, ibid. 9, No. 2, 405-448 (1997; Zbl 0905.11048)], whereby the structure of the Galois groups \(\text{Gal}(k_{\text{ur}}/k)\) of the maximal unramified extensions \(k_{\text{ur}}\) of imaginary quadratic number fields \(k\) of conductors less than 723 under the Generalized Riemann Hypothesis (GRH) was determined. In this paper the author improves upon this determination to conductors less than or equal to 1000, with the exception of four fields. In the process of obtaining the structure of \(\text{Gal}(k_{\text{ur}}/k)\), the significant result is obtained that for the field \(k=\mathbb{Q}(\sqrt{-856})\), \(k_{\text{ur}}=k_4\), the fourth Hilbert class field of \(k\). This is the first example of a number field whose class field tower has length four.
The techniques utilized in this paper are essentially similar to the techniques utilized in the author’s Part I (loc. cit.). For all \(k\) as above, it is demonstrated that \(k_{\text{ur}}=k_{\ell}\), where \(l\) is the length of the Hilbert class field tower of \(k\), and \(l=1, 2,3\), or 4. A useful strategy to establish this result involves showing that \([k_{\text{ur}}:k_{\ell}] < 60\), since \(|A_5|=60\) where \(A_5\) is the alternating group of degree 5, and \(A_5\) is the finite nonsolvable group of minimal order. Through utilizing lower bounds for the root discriminants of certain totally imaginary number fields, the author is able to establish that \(k_{\text{ur}}=k_{\ell}\) by showing that \(k_{\ell}\) does not have an unramified nonsolvable Galois extension. For many of the fields considered, this amounts to showing that \(k_{\ell}\) does not have an unramified \(A_5\)-extension which is normal over \(\mathbb{Q}\). For the cases of showing \(k_{\text{ur}}=k_{\ell}\) by using a general fact about the structure of group extensions of \(A_5\) and \(S_5\) (symmetric group of degree 5) by finite Abelian groups, the above criterion is reduced to showing the nonexistence of certain quintic number fields. This technique also enables the author to improve upon his table of \(\text{Gal}(k_{\text{ur}}/k)\) unconditionally (without GRH) from conductors \(\leq 420\) in his previous paper to conductors \(\leq 463\) (with the exception of conductor 427).
Through making use of results concerning 2-class field towers, CM-fields, relative class numbers, and genus theory, the author is able to demonstrate that there exist infinitely many imaginary quadratic number fields with 2-class field tower length \(\ell^{(2)} = 2\) and class field tower length \(\ell \geq 3\), as well as \(\ell^{(2)}=1\) and \(\ell\geq 3\). The above demonstration is related to the method of determining that for the field \(\mathbb{Q}(\sqrt{-984})\), \(k_{\text{ur}}\supseteq (k_g)_1\) and \(k_{\text{ur}}\neq (k_g)+1\), (\((k_g)_1\) is the Hilbert class field of the genus field \(k_g\) of \(k\)), there does not exist an \(S_4\)-extension \(M\) of \(\mathbb{Q}\) such that the composition \(kM\) is an unramified extension of \(k\) not contained in \((k_g)_1\), and there does exist a dihedral octic CM-field \(F\) such that the compositum \(kF_1\) is an unramified extension of \(k\) not contained in \((k_g)_1\)). The author notes that the degree of \(\mathbb{Q}(\sqrt{-984})\), which is 864, is the largest known degree of a number field with class number one (under GRH). Throughout the paper, the author makes extensive use of class number calculations utilizing KANT and PARI-GP.

MSC:
11R37 Class field theory
11R20 Other abelian and metabelian extensions
11R32 Galois theory
11R11 Quadratic extensions
Software:
KANT/KASH
PDF BibTeX XML Cite
Full Text: DOI EMIS Numdam EuDML
References:
[1] Benjamin, E., Lemmermeyer, F., Snyder, C., Imaginary quadratic fields k with cyclic Cl2(k1). J. Number Theory67 (1997), 229-245. · Zbl 0919.11074
[2] Gerth, F., III, The 4-class ranks of quadratic extensions of certain real quadratic fields. J. Number Theory33 (1989), 18-39. · Zbl 0694.12003
[3] Kisilevsky, H., Number fields with class number congruent to 4 mod 8 and Hilbert’s Theorem 94. J. Number Theory8 (1976), 271-279. · Zbl 0334.12019
[4] Lamacchia, S.E., Polynomials with Galois group PSL(2,7). Comm. Algebra8 (1980), 983-992. · Zbl 0436.12005
[5] Lefeuvre, Y., Louboutin, S., The class number one problem for dihedral CM-fields. In: Algebraic number theory and Diophantine analysis, F. Halter-Koch and R. F. Tichy eds. (Graz, 1998), de Gruyter, Berlin, 2000, 249-275. · Zbl 0958.11071
[6] Lemmermeyer, F., Ideal class groups of cyclotomic number fields. I. Acta Arith.72 (1998), 59-70. · Zbl 0901.11031
[7] Louboutin, S., The class number one problem for the dihedral and dicyclic CM-fields. Colloq. Math.80 (1999), 259-265. · Zbl 1036.11056
[8] Louboutin, S., Okazaki, R., Determination of all non-normal quartic CM-fields and of all non-abelian normal octic CM-fields with class number one. Acta Arith.67 (1994), 47-62. · Zbl 0809.11069
[9] Odlyzko, A.M., Some analytic estimates of class numbers and discriminants. Invent. Math.29 (1975), 275-286. · Zbl 0306.12005
[10] Odlyzko, A.M., Discriminant bounds, (unpublished tables), Nov. 29th 1976; available from http://www.dtc.umn.edu/odlyzko/unpublished/index.html
[11] Schur, I., Über die Darstellung der symmetrischen und alternierenden Gruppe durch gebrochene lineare substitutionen. J. Reine Angew. Math.139 (1911), 155-250. · JFM 42.0154.02
[12] Suzuki, M., Group theory. I Grundlehren der Mathematischen Wissenschaften 247, Springer-Verlag, Berlin-New York, 1982. · Zbl 0472.20001
[13] Uchida, K., Class numbers of imaginary abelian number fields. I. Tôhoku Math. J. (2) 23 (1971), 97-104. · Zbl 0213.06903
[14] Yamamoto, Y., Divisibility by 16 of class numbers of quadratic fields whose 2-class groups are cyclic. Osaka J. Math.21 (1984), 1-22. · Zbl 0535.12002
[15] Yamamura, K., Maximal unramified extensions of imaginary quadratic number fields of small conductors. J. Théor. Nombres Bordeaux9 (1997), 405-448. · Zbl 0905.11048
[16] Yamamura, K., Maximal unramified extensions of real quadratic number fields of small conductors, in preparation. · Zbl 1013.11076
[17] Yang, H.-S., Kwon, S.-H., The non-normal quartic CM-fields and the octic dihedral CM-fields with class number two. J. Number Theory79 (1999), 175-193. · Zbl 0976.11051
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.