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

Ken Yamamura

Journal de théorie des nombres de Bordeaux (2001)

  • Volume: 13, Issue: 2, page 633-649
  • ISSN: 1246-7405

Abstract

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In the previous paper [15], we determined the structure of the Galois groups Gal ( K u r / K ) of the maximal unramified extensions K u r of imaginary quadratic number fields K of conductors ≦ 1000 under the Generalized Riemann Hypothesis (GRH) except for 23 fields (these are of conductors ≧ 723 ) and give a table of Gal ( K u r / K ) . We update the table (under GRH). For 19 exceptional fields K of them, we determine Gal ( K u r / K ) . In particular, for K = 𝐐 ( - 856 ) , we obtain Gal ( K u r / K ) ≅ S 4 ˜ × C 5 and K u r = 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.

How to cite

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Yamamura, Ken. "Maximal unramified extensions of imaginary quadratic number fields of small conductors, II." Journal de théorie des nombres de Bordeaux 13.2 (2001): 633-649. <http://eudml.org/doc/248726>.

@article{Yamamura2001,
abstract = {In the previous paper [15], we determined the structure of the Galois groups $\text\{Gal\}(K_\{ur\}/K)$ of the maximal unramified extensions $K_\{ur\}$ of imaginary quadratic number fields $K$ of conductors $\leqq 1000$ under the Generalized Riemann Hypothesis (GRH) except for 23 fields (these are of conductors $\geqq 723$) and give a table of $\text\{Gal\}(K_\{ur\}/K)$. We update the table (under GRH). For 19 exceptional fields $K$ of them, we determine $\text\{Gal\}(K_\{ur\}/K)$. In particular, for $K = \mathbf \{Q\}(\sqrt\{-856\})$, we obtain $\text\{Gal\}(K_\{ur\}/K) \cong \widetilde\{S_4\} \times C_5 \text\{ and \} K_\{ur\} = K_4$, the fourth Hilbert class field of $K$. This is the first example of a number field whose class field tower has length four.},
author = {Yamamura, Ken},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {CM-field; root discriminant; Hilbert class field; -extension; nonsolvable Galois group; conductor},
language = {eng},
number = {2},
pages = {633-649},
publisher = {Université Bordeaux I},
title = {Maximal unramified extensions of imaginary quadratic number fields of small conductors, II},
url = {http://eudml.org/doc/248726},
volume = {13},
year = {2001},
}

TY - JOUR
AU - Yamamura, Ken
TI - Maximal unramified extensions of imaginary quadratic number fields of small conductors, II
JO - Journal de théorie des nombres de Bordeaux
PY - 2001
PB - Université Bordeaux I
VL - 13
IS - 2
SP - 633
EP - 649
AB - In the previous paper [15], we determined the structure of the Galois groups $\text{Gal}(K_{ur}/K)$ of the maximal unramified extensions $K_{ur}$ of imaginary quadratic number fields $K$ of conductors $\leqq 1000$ under the Generalized Riemann Hypothesis (GRH) except for 23 fields (these are of conductors $\geqq 723$) and give a table of $\text{Gal}(K_{ur}/K)$. We update the table (under GRH). For 19 exceptional fields $K$ of them, we determine $\text{Gal}(K_{ur}/K)$. In particular, for $K = \mathbf {Q}(\sqrt{-856})$, we obtain $\text{Gal}(K_{ur}/K) \cong \widetilde{S_4} \times C_5 \text{ and } K_{ur} = K_4$, the fourth Hilbert class field of $K$. This is the first example of a number field whose class field tower has length four.
LA - eng
KW - CM-field; root discriminant; Hilbert class field; -extension; nonsolvable Galois group; conductor
UR - http://eudml.org/doc/248726
ER -

References

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