\input amstex
\documentstyle{amsppt}
\define\Q{{\bold Q}}
\define\Gal{{\operatorname{Gal}}}
\define\Cl{{\operatorname{Cl}}}
\define\GL{{\operatorname{GL}}}
\define\SL{{\operatorname{SL}}}
\define\PGL{{\operatorname{PGL}}}
\define\PSL{{\operatorname{PSL}}}
\define\lt{\ltimes}
\define\rt{\rtimes}
\define\la{\langle}
\define\ra{\rangle}
\NoPageNumbers

\mag=\magstep1

\document
Let $K=\Q(\sqrt{d})$ be an imaginary quadratic number field
with discriminant $d$ and
$K_{\text{ur}}$ its maximal unramified extension.
We denote by $G$ the Galois group $\Gal(K_{\text{ur}}/K)$ and
$l$  be the length of the class field tower of $K$:
$K=K_0\subseteqq K_1\subseteqq K_2\subseteqq\cdots$
($K_{i+1}$ is the Hilbert class field of $K_i$),
that is, $l$ is the smallest number with $K_{l+1}=K_l$.

Under GRH we have determined the structure of $G$ for $|d|<1000$
except for $d=-883, -907, -947$, and $-996$.
For $|d|\leqq 467$, $d\ne-427$, the results are unconditional.
We give a table of $G$ for $K$ with $K_{\text{ur}}\ne K_1$ below.
For this table we omit the fields
$\Q(\sqrt{-883}), \Q(\sqrt{-907})$, and $\Q(\sqrt{-947})$,
for which we expect $K_{\text{ur}}=K_1$
because, if not, there wuold exist a $\PGL(2,7)$-extension of $\Q$
that is an unramified $\PSL(2,7)$-extenison of $K$,
however it is unlikely from experimental numerical data.

%\eightpoint
\NoBlackBoxes
\vskip.5cm
\def\hlf{\noalign{\hrule}}
\def\vspa{height2pt &\omit&&\omit&&\omit&&\omit&&\omit&&\omit&\cr}
\def\vspb
{height1pt &\omit&&\omit&&\omit&&\omit&&&\omit&&\omit&&\omit&&\omit&\cr}
\def\table#1{
\vbox\bgroup\offinterlineskip
\halign\bgroup
% preamble
\vrule## &
\ \hfil $##$ \ \strut & \vrule## & \ \hfil $##$ \hfil
& \vrule## & \ \hfil $##$ & \vrule## &\ \hfil $##$ \hfil & \vrule##\ &
\ \vrule## &
\ \hfil $##$ \ & \vrule## & \ \hfil $##$ \hfil
& \vrule## & \ \hfil $##$ & \vrule## &\ \hfil $##$ \hfil & \vrule##
\cr
%data
\multispan{18}{ \hfil #1 \hfil } \cr
\noalign{\smallskip}
\hlf
%\vspb
& -d && \Cl(K) && l && G & & & -d && \Cl(K) && l && G & \cr
%\vspb
\hlf
\vspb
}
\def\endtable{\egroup\egroup}
\NoBlackBoxes
\centerline{Table of imaginary quadratic number fields $K=\Q(\sqrt{d})$,
$|d|<1000$ with $K_{\text{ur}}\ne K_1$}
%\vskip-.5cm
$$
\table{}
& 115 && C_2 && 2 && D_3 &&
& 472 && C_6 && 2 && D_3\times C_3 & \cr
& 120 && V_4 && 2 && Q_8 &&
& 483 && V_4 && 2 && D_6 & \cr
& 155 && C_4 && 2 && Q_{12} &&
& 491 && C_9 && 3 && \widetilde{A_4}\curlywedge C_9 & \cr
& 184 && C_4 && 2 && Q_{12} &&
& 515 && C_6 && 2 && D_5\times C_3 & \cr
& 195 && V_4 && 2 && Q_{16} &&
& 520 && V_4 && 2 && Q_{24} & \cr
& 235 && C_2 && 2 && D_5 &&
& 527 && C_{18} && 2 && D_3\times C_9 & \cr
& 248 && C_8 && 2 && I_3^8 &&
& 535 && C_{14} && 2 && D_3\times C_7 & \cr
& 255 && C_6\times C_2 && 2 && Q_8\times C_3 &&
& 552 && C_4\times C_2 && 2 && Q_{12}\times C_2 & \cr
& 260 && C_4\times C_2 && 2 && M_{16} &&
& 555 && V_4 && 2 && Q_{32} & \cr
& 276 && C_4\times C_2 && 2 && Q_{12}\times C_2 &&
& 563 && C_9 && 3 && \widetilde{A_4}\curlywedge C_9 & \cr
& 280 && V_4 && 2 && Q_{16} &&
& 564 && C_4\times C_2 && 2 && Q_{20}\times C_2 & \cr
& 283 && C_3 && 3 && \widetilde{A_4} &&
& 568 && C_4 && 2 && Q_{28} & \cr
& 295 && C_8 && 2 && I_3^8 &&
& 580 && C_4\times C_2 && 2 && 32\Gamma_3f\lt C_3 & \cr
%& 580 && C_4\times C_2 && 2 && 32\Gamma_3f\curlywedge D_3 & \cr
& 299 && C_8 && 2 && I_3^8 &&
& 595 && V_4 && 2 && Q_{40} & \cr
& 312 && V_4 && 2 && Q_{16} &&
& 611 && C_{10} && 2 && D_5\times C_5 & \cr
& 331 && C_3 && 3 && \widetilde{A_4} &&
& 632 && C_{8} && 2 && I_5^8 & \cr
& 340 && V_4 && 2 && SD_{16} &&
& 635 && C_{10} && 2 && D_5\times C_5 & \cr
& 355 && C_4 && 2 && Q_{28} &&
& 643 && C_3 && 3 && \widetilde{A_4} & \cr
& 372 && V_4 && 2 && D_6 &&
& 644 && C_8\times C_2 && 2 && D_3\times C_8 & \cr
& 376 && C_8 && 2 && I_5^8 &&
& 651 && C_4\times C_2 && 2 && D_3\times C_4 & \cr
& 391 && C_{14} && 2 && D_3\times C_7 &&
& 655 && C_{12} && 2 && Q_{20}\times C_3 & \cr
& 395 && C_8 && 2 && I_5^8 &&
& 660 && C_2^3 && 2 && 64\Gamma_{15}f_2 & \cr
& 403 && C_2 && 2 && D_3 &&
& 663 && C_8\times C_2 && 2 && 64\Gamma_3p & \cr
& 408 && V_4 && 2 && D_4 &&
& 664 && C_{10} && 2 && D_3\times C_5 & \cr
& 415 && C_{10} && 2 && D_3\times C_5 &&
& 667 && C_4 && 2 && Q_{12} & \cr
& 420 && C_2^3 && 2 && 32\Gamma_4c_3 &&
& 680 && C_6\times C_2 && 2 && Q_{16}\times C_3 & \cr
& 435 && V_4 && 2 && Q_{16}\lt C_3 &&
%& 435 && V_4 && 2 && Q_{16}\curlywedge D_6 &&
& 687 && C_{12} && 3 && (S_4\curlywedge C_4)\times C_3 & \cr
& 440 && C_6\times C_2 && 2 && Q_{16}\times C_3 &&
& 695 && C_{24} && 2 && I^8_3\times C_3 & \cr
& 455 && C_{10}\times C_2 && 2 && Q_8\times C_5 &&
& 696 && C_6\times C_2 && 2 && Q_{24}\times C_3 & \cr
\hlf
\endtable
$$

\newpage

\vcorrection{-.35in}
$$
\table{Continued (under GRH)}
& 708 && V_4 && 2 && D_6 &&
& 856 && C_6 && 4 && \widetilde{S_4}\times C_3 & \cr
& 715 && V_4 && 2 && Q_{16}\lt C_5 &&
%& 715 && V_4 && 2 && Q_{16}\curlywedge D_5 &&
& 868 && C_4\times C_2 && 2 && Q_{12}\times C_2 & \cr
& 728 && C_6\times C_2 && 2 && Q_{16}\times C_3 &&
& 884 && C_8\times C_2 && 2 && 64\Gamma_3n_2 & \cr
& 731 && C_{12} && 3 && \widetilde{A_4}\curlyvee C_8 &&
& 888 && C_6\times C_2 && 2 && Q_{32}\times C_3 & \cr
& 740 && C_8\times C_2 && 2 && M_{32} &&
& 895 && C_{16} && 2 && I^{16}_5 & \cr
& 744 && C_6\times C_2 && 2 && D_3\times C_6 &&
& 899 && C_{14} && 2 && D_3\times C_7 & \cr
& 751 && C_{15} && 3 && \widetilde{A_4}\times C_{5} &&
& 915 && C_4\times C_2 && 2 && 64\Gamma_8e & \cr
& 755 && C_{12} && 2 && Q_{28}\times C_3 &&
& 916 && C_{10} &&  3 && S_4\times C_5 & \cr
& 759 && C_{12}\times C_2 && 2 && D_3\times C_{12} &&
& 920 && C_{10}\times C_2 && 2 && (Q_{16}\lt C_3)\times C_5 & \cr
%& 920 && C_{10}\times C_2 && 2 && (Q_{16}\curlywedge D_6)\times C_5 & \cr
& 760 && V_4 && 2 && Q_{32}\lt C_3 &&
%& 760 && V_4 && 2 && Q_{32}\curlywedge D_6 &&
& 923 && C_{10} && 2 && D_7\times C_5 & \cr
& 767 && C_{22} && 2 && D_3\times C_{11} &&
& 935 && C_{14}\times C_2 && 2 && Q_{16}\times C_7 & \cr
& 771 && C_6 && 3 && S_4\times C_3 &&
& 943 && C_{16} && 2 && I^{16}_3 & \cr
& 795 && V_4 && 2 && Q_{40} &&
& 948 && C_6\times C_2 &&2 && D_5\times C_6 & \cr
& 799 && C_{16} && 2 && I^{16}_{5} &&
& 952 && C_4\times C_2 && 2 && (C_4\wr C_2)\lt C_5 & \cr
%& 952 && C_4\times C_2 && 2 && (C_4\wr C_2)\curlywedge D_5 & \cr
& 820 && C_4\times C_2 && 2 && 64\Gamma_8c_2 &&
& 955 && C_4 && 2 && Q_{52} & \cr
& 824 && C_{20} && 2 && Q_{20}\times C_5 &&
& 984 && C_6\times C_2 && 3 && (Q_{16}\lt C_3^2)\times C_3 & \cr
& 835 && C_6 && 2 && D_{11}\times C_3 &&
& 987 && C_4\times C_2 && 2 && Q_{20}\times C_2 & \cr
& 840 && C_2^3 && 2 && 32\Gamma_4d &&
& 995 && C_8 && 2 && I_9^8 & \cr
& 851 && C_{10} && 2 && D_3\times C_5 &&
& 996 && C_6\times C_2 && 3\text{ or }4 && ? & \cr
& 852 && C_4\times C_2 && 2 && Q_{28}\times C_2 &&
&  &&  &&  &&  & \cr
\hlf
\endtable
\hfil
$$

The fields with $|d|<1000$, $d\ne -883, -907, -947$
can be classified simply as follows:
\eightpoint
\NoBlackBoxes
$$
\matrix
	\format \l & \l & \l \\
	\matrix
		\format \l \\
		\noalign{\vskip22pt}
		\left\{
		\matrix
		\\
		\noalign{\vskip28pt}
		\\
		\endmatrix
		\right. 
	\endmatrix &
	\hskip-5pt
	\matrix
		\format \l \\
		\noalign{\vskip22pt}
		\hskip-1pt
		d_E\nmid d{\hskip1pt}\Bigg\{ \\
		\\
		\noalign{\vskip9pt}
		\\
		d_E\mid d{\hskip1pt}\Bigg\{
	\endmatrix &
	\hskip-3pt
	\matrix
		\format \l & \l & \l \\
		\matrix
			\format \l & \l \\
			\matrix
				\format \l \\
				\noalign{\vskip5pt}
				\hskip-1pt
				d\ne-856,-984,-996\biggm\{ \\
			\endmatrix &
			\hskip-1pt
			\matrix
				\format \l & \l \\
				h(K_{\text{gen}})
				=\frac{h(K)}{[K_{\text{gen}}:K]} \biggr\{ &
				\matrix
					\format \l \\
					h(K)=1 \\
					\noalign{\smallskip}
					h(K)>1
				\endmatrix \\
				h(K_{\text{gen}})
				>\frac{h(K)}{[K_{\text{gen}}:K]} &
			\endmatrix
		\endmatrix &
		\hskip-5pt
		\left.
		\matrix
			\format \l & \c \\
			\cdots & K_{\text{ur}}=K \\
			\noalign{\smallskip}
			\cdots & K_{\text{ur}}=K_1 \\
			\noalign{\smallskip}
			\cdots & K_{\text{ur}}=K_2 \\
		\endmatrix
		\right\} &
		K_{\text{ur}}=(K_{\text{gen}})_1 \\
		\noalign{\smallskip}
		\matrix
			\format \l \\
			d=-856,-984,-996 \\
			\noalign{\smallskip}
			\matrix
				\format \l & \l \\
				\hskip-1pt
				d=d_E \Bigm\{ &
				\matrix
					\format \l \\
					-\text{\,prime}\\
					\noalign{\smallskip}
					\text{\,composite}
				\endmatrix
			\endmatrix \\
			\noalign{\smallskip}
			d=d'd_E\ 
			(d'\text{\,fundamental quadratic discriminant})
		\endmatrix &
		\hskip-5pt
		\left.
		\matrix
			\format \l & \c \\
			\cdots & l\geqq 3 \\
			\noalign{\smallskip}
			\cdots & K_{\text{ur}}=K_3 \\
			\noalign{\smallskip}
			\cdots & l\geqq 2 \\
			\noalign{\smallskip}
			\cdots & l\geqq 3 \\
		\endmatrix
		\right\} &
		K_{\text{ur}}\supsetneqq (K_{\text{gen}})_1
	\endmatrix
\endmatrix
$$
\tenpoint
Here, we denote by $K_{\text{gen}}$ the genus field of $K$
and by $(K_{\text{gen}})_1$ its Hilbert class field.
We mean by `$d_E\mid d$' that the discriminant $d$ of $K$ is
divisible by the discriminant of some quartic number field $E$.
As a remarkable observation, we note that
for most $K$ with $|d|<1000$ we have $K_{\text{ur}}=(K_{\text{gen}})_1$.
In fact, among $305$ such fields, at most $16$ are exceptional.
Why is the degree $[K_{\text{ur}}:(K_{\text{gen}})_1]$ small ?
We may expect that $[K_{\text{ur}}:(K_{\text{gen}})_1]$ is small
for fields we consider on the ground of the following.

\proclaim{Proposition 1} 
Let $L$ be the Hilbert class field of the gunus field $K_{\text{gen}}$ of
an imaginary abelian number field $K$.
Then for any prime number $p$ with $p\nmid [L:\Q]$,
the $p$-class group $\Cl^{(p)}(L)$ of $L$ is trivial or noncyclic.
\endproclaim

For all $K$ with $|d|<1000$
such that $h(K_{\text{gen}})>h(K)/[K_{\text{gen}}:K]$,
which is equivalent to $(K_{\text{gen}})_1\supsetneqq K_1$,
we have $K_2=(K_{\text{gen}})_1$. %, that is,
%the second Hilbert class field of $K$ coincides with 
%the Hilbert class field of the genus field $K_{\text{gen}}$ of $K$.
Thus, the following problem arises naturally:

\proclaim{Problem 1} 
Characterize the imaginary quadratic number fields $K$
with $K_2=(K_{\text{gen}})_1\ne K_1$.
\endproclaim

\noindent
For $h(K_{\text{gen}})>h(K)/[K_{\text{gen}}:K]$,
$h(K)$ must necessarily be even and
$h(K)$ is even if and only if $d$ has (at least) two distinct prime factors.
Probably, for most $K$, this inequality holds.
In fact, if a quadratic subfield $\ne K$ of $K_{\text{gen}}$
has class number divisible by an odd prime $p$,
then we have $h(K_{\text{gen}})\geqq ph(K)/[K_{\text{gen}}:K]$.
However, the characterization of such $K$ is very difficult.
For example, let consider $K=\Q(\sqrt{-q})$
with $q$ a prime $\equiv 1\pmod{4}$.
Then we have $h(K_{\text{gen}})=h(\Q(\sqrt{q}))h(K)/[K_{\text{gen}}:K]$
and therefore the inequality holds if and only if $h(\Q(\sqrt{q}))>1$.
Moreover the characterization of $K$ with $K_2=(K_{\text{gen}})_1$
is also difficult.
But we can easily find $K$
with $K_2\supsetneqq(K_{\text{gen}})_1\supsetneqq K_1$.
The inclusion $K_2\supsetneqq(K_{\text{gen}})_1$ holds
for $K$ whose $p$-class group has rank $\geqq 2$.
For such $K$, the length of the $p$-class field tower of $K$:
$K=K_0^{(p)}\subseteqq K_1^{(p)}\subseteqq K_2^{(p)}\subseteqq\cdots$
($K_{i+1}^{(p)}$ is the Hilbert class field of $K_i^{(p)}$)
is larger than one, that is, $K_2^{(p)}\ne K_1^{(p)}$.
Since $K_2^{(p)}$ and $K_{\text{gen}}$ is linearly independent over $K$,
we have
$K_2\supseteqq (K_{\text{gen}})_1K_2^{(p)}\supsetneqq (K_{\text{gen}})_1$.
Hence if moreover $(K_{\text{gen}})_1\ne K_1$,
we have $K_2\supsetneqq(K_{\text{gen}})_1\supsetneqq K_1$.
For example, let $K=\Q(\sqrt{-3896})$.
Then we have $\Cl(K)\cong C_{12}\times C_3,
K_{\text{gen}}=\Q(\sqrt{2},\sqrt{-487})$,
and $\Cl(K_{\text{gen}})\cong C_{42}\times C_3$.
Therefore
$K_2\supsetneqq K_1^{(3)}(K_{\text{gen}})_1=(K_{\text{gen}})_1\supsetneqq K_1$.
Consequently, for $K_2=(K_{\text{gen}})_1$
it is necessary that the odd part of the class group of $K$ is cyclic.
Thus, it is reasonable to consider the following
before treating Problem 1.

\proclaim{Problem 2} 
Characterize the imaginary quadratic number fields $K$ with
$$
K_2^{(2)}=(K_{\text{gen}})_1^{(2)}\ne K_1^{(2)}.
$$
\endproclaim

\noindent
By known results on the $2$-class field tower,
if the $2$-class group of $K_1^{(2)}$ is nontrivial and cyclic,
$K_2^{(2)}=(K_{\text{gen}})_1^{(2)}\ne K_1^{(2)}$ holds.
No example of $K$
with $K_2^{(2)}\supsetneqq (K_{\text{gen}})_1^{(2)}\supsetneqq K_1^{(2)}$ is known.
\medpagebreak

For $K$ satisfying the stronger condition
$E_1=K_1$ for all intermediate fields $E$ of $K_1/K$
than the condition $(K_{\text{gen}})_1=K_1$,
we can estimate $h(K_1)$ from below in the case $h(K_1)>1$
by considering the action of Galois groups on class groups.

\proclaim{Proposition 2} 
Let $K$ be an imaginary quadratic number field
with cyclic class group of odd order $n\geqq 7$.
Assume that for all intermediate fields $E$ of $K_1/K$, we have $E_1=K_1$.
Then if $h(K_1)>1$ {\rm(}this is equivalent to $K_2\ne K_1${\rm)},
we have $h(K_1)\geqq 2^6=64$.
{\rm(}This is valid also for real quadratic number fields.
In the case where $K$ is imaginary and $h(K)$ is a prime {\rm(}$\geqq 7${\rm)},
$h(K_1) \geqq 64$ can be improved to $h(K_1)\geqq 13^2=169$.
Moreover, in the case where $h(K)\geqq 11$ and $h(K)\ne 21, 63$,
$h(K_1)\geqq 64$ can be improved to $h(K_1)\geqq 2^8=256$
{\rm(}without assuming the imaginarity of $K$.{\rm)}
{\rm)}
\endproclaim

\noindent
The first known imaginary quadratic number field
with $h(K)=q\geqq 7$ ($q$\ an odd prime) and $l\geqq 2$
is $\Q(\sqrt{-14947})$.
(For $K=\Q(\sqrt{-14947})$, we have $h(K)=17$ and $256\mid h(K_1)$.)
It is thought likely that this field is
really the first such field.

\proclaim{Proposition 3} 
Let $K$ be an imaginary quadratic number field
with nontrivial cyclic $2$-class group.
\newline
{\rm (i)} Assume $\Cl(K)\cong C_{2^m} (m\geqq 3)$ and
that the unique unramified cyclic quartic extension $E$ of $K$
has class number $2^{m-2}$, which implies $h(K_{\text{gen}})=h(K)/2$.
If $h(K_1)>1$, then we have $h(K_1)\geqq 7^2=49$.
\newline
{\rm (ii)} Assume $h(K)=2q$ {\rm(}$q$\ an odd prime $\geqq 7${\rm)}
and $h(K_{\text{gen}})=q$ $(=h(K)/2)$.
If $h(K_1)>1$, then we have $h(K_1)\geqq 2^6=64$.
\newline
{\rm(}These are valid also for real quadratic number fields
if the wide genus field $K_{\text{gen}}$ is
the unique unramified quadratic extension of $K$.{\rm)}
\endproclaim

\proclaim{Question} 
What is the first imaginary quadratic number field
having an unramified nonsolvable Galois extension ?
{\rm(}What is the first $K$ with $K_{ur}\ne K_l$ ?{\rm)}
\endproclaim

\noindent
Recent data for quintic number fields enable us to
give a partial answer:

\proclaim{Proposition 4}
The field $\Q(\sqrt{-1507})$ is the first imaginary quadratic number field
having an unramified $A_5$-extension which is normal over $\Q$
in the sense that none of $\Q(\sqrt{d})$ of discriminant $d$
with $0>d>-1507$ has such an extension.
Moreover, such an extension of $\Q(\sqrt{-1507})$ is
given by the composite field of it with the splitting field of
the quintic polynomial $X^5-5X^3+5X^2+24X+4$,
which is an $A_5$-extension of $\Q$.
\endproclaim

\enddocument

Now we explain the notations in our table.
As usual, $C_n$ is the cyclic group of order $n$,
$V_4$ is the four group, that is, $V_4=C_2^2=C_2\times C_2$,
$D_n$ $(n\geqq 3)$ is the dihedral group of order $2n$,
$Q_{4n}$ $(n\geqq 2)$ is the generalized quaternion group of order $4n$,
and $SD_{8n}$ $(n\geqq 2)$ is the semi-dihedral group of order $8n$:
$$
\aligned
D_n&=\la\,a,b\,\mid\,a^n=b^2=1,\ b^{-1}ab=a^{-1}\,\ra,\\
Q_{4n}&=\la\,a,b\,\mid\,a^{2n}=1,\ b^2=a^n,\ b^{-1}ab=a^{-1}\,\ra,\\
SD_{8n}&=\la\,a,b\,\mid\,a^{4n}=b^2=1,\ b^{-1}ab=a^{2n-1}\,\ra.
\endaligned
$$

\noindent
$I_n^{2m}$ $(m\geqq 2, n\geqq 3)$ denotes the group of order $2mn$ given by
$$
\la\,a,b\,\mid\,a^{2m}=b^n=1,\ a^{-1}ba=b^{-1}\,\ra.
$$
$M_{2^n}$ $(n\geqq 4)$
denotes the modular group of order $2^n$ given by
$$
\la\,a,b\,\mid\,a^{2^{n-1}}=b^2=1,\ b^{-1}ab=a^{2^{n-2}+1}\,\ra.
$$
$\widetilde{A_4}$ is the double cover of $A_4$\,:\,$\widetilde{A_4}\cong\SL(2,3)$.

\noindent
For some $2$-groups
we use designations given in the table due to M. Hall and J. K. Senior.
\medskip

