\input amstex
\documentstyle{amsppt}
\define\Q{{\bold Q}}
\define\Gal{{\operatorname{Gal}}}
\define\PSL{{\operatorname{PSL}}}
\define\PGL{{\operatorname{PGL}}}
\define\Cl{{\operatorname{Cl}}}
\define\lt{\ltimes}
\define\rt{\rtimes}
\NoPageNumbers

\mag=\magstep1

\document
Let $K=\Q(\sqrt{d})$ be a real quadratic number field with discriminant $d$
and $K_{\text{w-ur}}$ its maximal weakly unramified extension.
We denote by $\Gamma$ the Galois group $\Gal(K_{\text{w-ur}}/K)$ and
$l_w$ be the length of the narrow Hilbert class field tower of $K$:
$K=K_{0,\text{nar}}\subseteqq K_{1,\text{nar}}\subseteqq
K_{2,\text{nar}}\subseteqq\cdots$
($K_{i+1,\text{nar}}$ is the Hilbert class field of $K_{i,\text{nar}}$),
that is, $l_w$ is the smallest number
with $K_{l_w+1,\text{nar}}=K_{l_w,\text{nar}}$.

Under GRH we have determined the structure of $\Gamma$ for $d\leqq 800$
except for $d=641, 661, 673, 677, 701, 709, 757, 769, 773, 776, 797$.
%For $d\leqq 2636$, the results are unconditional.
We give a table of $\Gamma$ for $K$
for which we suceed in concluding $K_{\text{w-ur}}\ne K_{1,\text{nar}}$ below,
however we omit $K$ for which we expect $K_{\text{w-ur}}=K$
but we have not suceeded in proving yet.


\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_w && \Gamma & & & d && \Cl(K) && l_w && \Gamma & \cr
%\vspb
\hlf
\vspb
}
\def\endtable{\egroup\egroup}
\NoBlackBoxes
\centerline{Table of real quadratic number fields $K=\Q(\sqrt{d})$,
$d\leqq 800$ with $K_{\text{w-ur}}\ne K_{1,\text{nar}}$}
%\vskip-.5cm
$$
\table{}
&  69 && C_2 && 2 && D_3 &&
& 357 && V_4 && 2 && D_{10} & \cr
&  92 && C_2 && 2 && D_3 &&
& 376 && C_2 && 2 && D_5 & \cr
&  93 && C_2 && 2 && D_3 &&
& 380 && V_4 && 2 && D_8 & \cr
& 124 && C_2 && 2 && D_3 &&
& 381 && C_2 && 2 && D_5 & \cr
& 141 && C_2 && 2 && D_5 &&
& 385 && V_4 && 2 && D_4 & \cr
& 156 && V_4 && 2 && D_4 &&
& 393 && C_2 && 2 && D_5 & \cr
& 161 && C_2 && 2 && D_3 &&
& 408 && V_4 && 2 && D_4 & \cr
& 165 && V_4 && 2 && D_4 &&
& 412 && C_2 && 2 && D_5 & \cr
& 168 && V_4 && 2 && D_4 &&
& 413 && C_2 && 2 && D_3 & \cr
& 177 && C_2 && 2 && D_3 &&
& 417 && C_2 && 2 && D_3 & \cr
& 184 && C_2 && 2 && D_3 &&
& 428 && C_2 && 2 && D_3 & \cr
& 188 && C_2 && 2 && D_5 &&
& 429 && V_4 && 2 && D_4\curlywedge D_5 & \cr
& 204 && V_4 && 2 && D_4 &&
& 437 && C_2 && 3 && S_4 & \cr
& 213 && C_2 && 2 && D_7 &&
& 440 && V_4 && 2 && D_4 & \cr
& 217 && C_2 && 2 && D_3 &&
& 444 && V_4 && 2 && D_8 & \cr
& 220 && V_4 && 2 && D_4 &&
& 453 && C_2 && 2 && D_7 & \cr
& 229 && C_3 && 2 && A_4 &&
& 456 && V_4 && 2 && D_6 & \cr
& 236 && C_2 && 2 && D_3 &&
& 460 && V_4 && 2 && D_6 & \cr
& 237 && C_2 && 2 && D_5 &&
& 465 && V_4 && 2 && D_4\curlywedge D_6 & \cr
& 248 && C_2 && 2 && D_3 &&
& 472 && C_2 && 3 && S_4 & \cr
& 249 && C_2 && 2 && D_3 &&
& 476 && V_4 && 2 && D_4\curlywedge D_{10} & \cr
& 253 && C_2 && 2 && D_3 &&
& 492 && V_4 && 2 && D_8 & \cr
& 257 && C_3 && 2 && A_4 &&
& 497 && C_2 && 2 && D_7 & \cr
& 273 && V_4 && 2 && D_4 &&
& 501 && C_2 && 2 && D_{11} & \cr
& 284 && C_2 && 2 && D_7 &&
& 508 && C_2 && 2 && D_5 & \cr
& 285 && V_4 && 2 && D_8 &&
& 517 && C_2 && 2 && D_5 & \cr
& 309 && C_2 && 2 && D_5 &&
& 520 && V_4 && 2 && Q_8 & \cr
& 312 && V_4 && 2 && D_{12} &&
& 524 && C_2 && 2 && D_5 & \cr
& 316 && C_6 && 2 && D_5\times C_3 &&
& 537 && C_2 && 2 && D_5 & \cr
& 321 && C_6 && 2 && D_3\times C_3 &&
& 552 && V_4 && 2 && Q_{12} & \cr
& 329 && C_2 && 2 && D_5 &&
& 553 && C_2 && 2 && D_5 & \cr
& 332 && C_2 && 3 && S_4 &&
& 556 && C_2 && 2 && D_3 & \cr
& 341 && C_2 && 3 && S_4 &&
& 568 && C_6 && 2 && D_7\times C_3 & \cr
& 345 && V_4 && 2 && D_6 &&
& 572 && V_4 && 2 && D_{10} & \cr
& 348 && V_4 && 2 && D_3\times D_3 &&
& 573 && C_2 && 2 && D_{13} & \cr
\hlf
\endtable
\hfil
$$

\newpage

$$
\table{Continued (under GRH)}
& 581 && C_2 && 2 && D_3 &&
& 705 && V_4 && 2 && D_{10} & \cr
& 589 && C_2 && 2 && D_3 &&
& 713 && C_2 && 2 && D_3\curlywedge D_3 & \cr
& 597 && C_2 && 3 && S_4\curlywedge D_9 &&
& 716 && C_2 && 2 && D_5 & \cr
& 604 && C_2 && 2 && D_7 &&
& 717 && C_2 && 3 && S_4\curlywedge D_5 & \cr
& 609 && V_4 && 2 && D_4\curlywedge D_6 &&
& 721 && C_2 && 2 && D_5 & \cr
& 616 && V_4 && 2 && D_4 &&
& 728 && V_4 && 2 && D_6 & \cr
& 620 && V_4 && 2 && D_4\curlywedge D_6 &&
& 732 && V_4 && 2 && D_8\curlywedge D_6 & \cr
& 632 && C_2 && 2 && D_5 &&
& 741 && V_4 && 2 && D_4\curlywedge D_6 & \cr
& 633 && C_2 && 2 && D_3 &&
& 744 && V_4 && 2 && D_8\lt C_3 & \cr
& 636 && V_4 && 2 && D_5\times D_3 &&
& 748 && V_4 && 2 && D_4 & \cr
& 645 && V_4 &&  2 && D_{14} &&
& 749 && C_2 && 4 && \widetilde{S_4} & \cr
& 649 && C_2 && 2 && D_3 &&
& 753 && C_2 && 2 && D_7 & \cr
& 653 && \{1\} && 0 && A_5 &&
& 760 && V_4 && 2 && D_8 & \cr
& 664 && C_2 && 2 && D_3 &&
& 761 && C_3 && 2 && A_4 & \cr
& 665 && V_4 && 2 && D_8 &&
& 764 && C_2 && 2 && D_{13} & \cr
& 668 && C_2 && 2 && D_{11} &&
& 776 && C_4 && \geqq 2 && C_4\lt C_3^2? & \cr
& 669 && C_2 && 2 && D_7 &&
& 777 && C_4\times C_2 && 2 && (C_4\times C_2)\lt C_8 & \cr
& 680 && V_4 && 2 && D_4 &&
& 780 && C_2^3 && 2 && C_2^3\lt C_4^2 & \cr
& 681 && C_2 && 2 && D_5 &&
& 781 && C_2 && 2 && D_7 & \cr
& 696 && V_4 && 2 && D_6 &&
& 789 && C_2 && 2 && D_{13} & \cr
& 697 && C_6 && 2 && A_4\times C_2 &&
& 796 && C_2 && 2 && D_9 & \cr
\hlf
\endtable
\hfil
$$

As a remarkable fact, we note the following.
For $K=\Q(\sqrt{653})$, we have $\Gal(K_{\text{w-ur}}/K)\cong A_5$,
a nonabelian simple group.
For this $K$ we have $\Gal(K_{\text{w-ur}}/\Q)$
\newline
$\cong A_5\times C_2$.
Though this is conditional,
this is only example of a number field $K$
for which $\Gal(K_{\text{w-ur}}/K)$ is nonabelian simple.
In the range $d\leqq 2001$,
for which we have $[K_{\text{w-ur}}:K]<\infty$ under GRH,
another fields have weakly unramified nonsolvable Galois extension:
$\Q(\sqrt{1609})$ has a weakly unramified $A_5$-extension
that is an $S_5$-extension of $\Q$.
$\Q(\sqrt{1417})$ has a weakly unramified $\PSL(2, 7)$-extension
that is an $\PSL(2,7)\times C_2$-extension of $\Q$.
$\Q(\sqrt{1709})$ has a weakly unramified $\PSL(2, 7)$-extension
that is an $\PGL(2,7)$-extension of $\Q$.

\enddocument

We the following notations in the table.
In the simple expressions of $K_1$ and $K_2$,
$a_i, b_i$ and $c_i$ denote any algebraic numbers generating
the $i$th totally real cubic number field,
the $i$th totally real quartic number field
with Galois group isomorphic to $S_4$, and
the $i$th totally real quintic number field
with Galois group isomorphic to $D_5$, respectively,
where we consider that the number fields
of each signature and each type (of Galois group of normal closure)
are numbered up to conjugacy by absolute values of discriminants.
(Here we do not need to consider nonisomorphic fields with same discriminants.)

