Maximal unramified extensions of imaginary quadratic number fields of small conductors. II.

*(English)*Zbl 1013.11076This 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.

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.

Reviewer: E.Benjamin (Belfast/Maine)

##### MSC:

11R37 | Class field theory |

11R20 | Other abelian and metabelian extensions |

11R32 | Galois theory |

11R11 | Quadratic extensions |

##### Keywords:

CM-field; root discriminant; Hilbert class field; \(A_5\)-extension; nonsolvable Galois group; conductor##### Software:

KANT/KASH##### References:

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