\input amstex
\documentstyle{amsppt}

\define\Q{{\bold Q}}
\define\Z{{\bold Z}}
\define\Gal{{\operatorname{Gal}}}
\define\SL{{\operatorname{SL}}}

\mag=\magstep1
\topmatter

\title List of known nonabelian normal CM-fields with class number one
\endtitle

\rightheadtext{nonabelian normal CM-fields with class number one}

%\author Ken Yamamura \endauthor

\endtopmatter

\document

A. M. Odlyzko proved that
there exist only finitely many normal CM-fields
with (relative) class number one.
J. Hoffstein proved that their degrees are at most $432$,
and later S. Bessassi improved this to $216$ ($96$ under GRH).
K. Yamamura completed the determination of
the imaginary abelian number fields with class number one,
and later Y.-H. Park and S.-H. Kwon determined
the imaginary abelian number fields
with relative class number one and class number $>1$.
However, the nonabelian normal CM-fields with class number one
have not been determined comletely yet.

Up to now (February 2008),
the normal CM-fields
with (relative) class number one of degrees $n$ with $n<48$,
have been determined
and also some types of such fields of degree $48$
have been determined.
The numbers of such nonabelian CM-fields with Galois group $G$
are as follows:
($\#_1$ (resp. $\#_2$) denotes the number of fields
with class number one
(resp. with relative class number one and class number $>1$)
and $\#=\#_1+\#_2$.)
$$
\vbox{
\offinterlineskip
% preamble
\halign{
\strut\vrule#&
\ \hfil $#$ \ & \vrule# & \ \hfil $#$ \ \hfil & \vrule# &
\ \hfil $#$ \ & \vrule# & \ \hfil $#$ \ & \vrule# & \ \hfil $#$ \ &
\vrule# \cr
\noalign{\smallskip}
\noalign{\hrule}
& n && G && \#_1 && \#_2 && \# & \cr
\noalign{\hrule}
&  8 && D_4 && 17 && 2 && 19 & \cr
& 12 && D_6 &&  9 && 7 && 16 & \cr
& 16 && Q_8\times C_2 &&  1 && 0 &&  1 & \cr
& 16 && D_4\times C_2 &&  4 && 0 &&  4 & \cr
& 16 && D_4\curlywedge C_4 &&  2 && 0 &&  2 & \cr
& 16 && D_8 &&  4 && 1 &&  5 & \cr
& 16 && D_4\curlyvee C_4 &&  1 && 0 &&  1 & \cr
& 20 && D_{10} &&  1 && 1 &&  2 & \cr
& 24 && D_{12} &&  1 && 0 &&  1 & \cr
& 24 && A_4\times C_2 &&  2 && 0 &&  2 & \cr
& 24 && \SL(2,3) && 1 && 0 &&  1 & \cr
& 24 && D_6\times C_2 &&  1 && 0 &&  1 & \cr
& 24 && D_3\times C_4 &&  1 && 0 &&  1 & \cr
& 24 && D_4\times C_3 &&  1 && 0 &&  1 & \cr
& 32 && (Q_8\curlywedge C_4)\times C_2 &&  1 && 0 &&  1 & \cr
& 32 && 32\Gamma_2d &&  1 && 0 &&  1 & \cr
& 32 && 32\Gamma_2j_1 &&  1 && 0 &&  1 & \cr
& 32 && M_{16}\times C_2 &&  3 && 0 &&  3 & \cr
& 36 && D_6\times C_3 &&  3 && 0 &&  3 & \cr
& 40 && F_{20}\times C_2 &&  1 && 0 &&  1 & \cr
& 48 && D_8\rtimes C_3 &&  1 && 0 &&  1 & \cr
\noalign{\hrule}
}}
$$
\newpage

\noindent
As usual, $C_n$ is the cyclic group of order $n$,
$D_n$ $(n\geqq 3)$ is the dihedral group of order $2n$,
$Q_8$ is the quaternion group,
$F_n$ is the Frobenius group of order $n$, and
$A_4$ is the alternating group of degree four.
For nonabelian groups $G$ of order less than $32$ that are not listed,
ther exist no normal CM-fields with Galois group $G$.

We list here known such fields.

We use the following notations.
Let $K$ be a nonabelian normal CM-field with relative class number one,
and $K_+$ its maximal totally real subfield.
We denote by $d_K, rd_K$ and $n$
the discriminant, root discriminant, and degree of $K$, respectively.
We put $G=\Gal(K/\Q)$.

\newpage

(I) $n=8$ ($17$ fields).
\medpagebreak

S. Louboutin and R. Okazaki determined
the nonabelian normal octic CM-fields with class number one.
There are exactly $17$ such fields and they are all dihedral.

\NoBlackBoxes
\def\hlf{\noalign{\hrule}}
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\def\vspb{height.5pt&\omit&&\omit&&\omit&&\omit& \cr}
\def\tableA#1{
\vbox\bgroup\offinterlineskip
\halign\bgroup
 \vrule##&\ $##$ \ \hfil &
 \vrule##&\ $##$ \ \hfil &
 \vrule##&\strut\ $##$ \hfil &
 \vrule##&\ \hfil $##$ \ &
 \vrule## \cr                            % preamble
\multispan{8} \hfil {#1} \hfil \cr
\noalign{\vskip 0.3cm}
\hlf                                         % full hline
\vspa                                        % wide 2pt
\vspb
& \multispan{1} \hfil {$K$} \hfil &&
\multispan{1} \hfil {$K_+$} \hfil &&
\multispan{1} \hfil {$d_K$}  \hfil &&
rd_K & \cr
\vspb                                        % wide 1pt
\hlf
\vspb
}
\def\endtable{\egroup\egroup}
$$
\tableA{$G=D_4$ ($17$ fields).}
& K_+(\sqrt{\smash{-(5+2\sqrt{2})}\vphantom{-3+4\sqrt{2}}}) &&
\Q(\sqrt{2},\sqrt{17}) && 2^{12}\cdot 17^4 && 11.661 & \cr
\hlf\vspb
& K_+(\sqrt{\smash{-(13+\sqrt{5})/2}\vphantom{-3+4\sqrt{2}}}) &&
\Q(\sqrt{5},\sqrt{41}) && 5^4\cdot 41^4 && 14.317 & \cr
\hlf\vspb
& K_+(\sqrt{\smash{-(9+\sqrt{13})/2}\vphantom{-3+4\sqrt{2}}}) &&
\Q(\sqrt{13},\sqrt{17}) && 13^4\cdot 17^4 && 14.866 & \cr
\hlf\vspb
& K_+(\sqrt{\smash{-(17+3\sqrt{5})/2}\vphantom{-3+4\sqrt{2}}}) &&
\Q(\sqrt{5},\sqrt{61}) && 5^4\cdot 61^4 && 17.464 & \cr
\hlf\vspb
& K_+(\sqrt{\smash{-(9+2\sqrt{13})}\vphantom{-3+4\sqrt{2}}}) &&
\Q(\sqrt{13},\sqrt{29}) && 13^4\cdot 29^4 && 19.416 & \cr
\hlf\vspb
& K_+(\sqrt{\smash{-(21+\sqrt{5})/2}\vphantom{-3+4\sqrt{2}}}) &&
\Q(\sqrt{5},\sqrt{109}) && 5^4\cdot 109^4 && 23.345 & \cr
\hlf\vspb
& K_+(\sqrt{\smash{-(9+2\sqrt{2})}\vphantom{-3+4\sqrt{2}}}) &&
\Q(\sqrt{2},\sqrt{73}) && 2^{12}\cdot 73^4 && 24.166 & \cr
\hlf\vspb
& K_+(\sqrt{\smash{-(11+4\sqrt{2})}\vphantom{-3+4\sqrt{2}}}) &&
\Q(\sqrt{2},\sqrt{89}) && 2^{12}\cdot 89^4 && 26.683 & \cr
\hlf\vspb
& K_+(\sqrt{\smash{-(13+2\sqrt{5})}\vphantom{-3+4\sqrt{2}}}) &&
\Q(\sqrt{5},\sqrt{149}) && 5^4\cdot 149^4 && 27.294 & \cr
\hlf\vspb
& K_+(\sqrt{\smash{-(13+2\sqrt{29})}\vphantom{-3+4\sqrt{2}}}) &&
\Q(\sqrt{29},\sqrt{53}) && 29^4\cdot 53^4 && 39.204 & \cr
\hlf\vspb
& K_+(\sqrt{\smash{-(19+\sqrt{233})/2}\vphantom{-3+4\sqrt{2}}}) &&
\Q(\sqrt{2},\sqrt{233}) && 2^{12}\cdot 233^4 && 43.174 & \cr
\hlf\vspb
& K_+(\sqrt{\smash{-(41+5\sqrt{5})/2}\vphantom{-3+4\sqrt{2}}}) &&
\Q(\sqrt{5},\sqrt{389}) && 5^4\cdot 389^4 && 44.102 & \cr
\hlf\vspb
& K_+(\sqrt{\smash{-(25+6\sqrt{13})}\vphantom{-3+4\sqrt{2}}}) &&
\Q(\sqrt{13},\sqrt{157}) && 13^4\cdot 157^4 && 45.177 & \cr
\hlf\vspb
& K_+(\sqrt{\smash{-(17+2\sqrt{2})}\vphantom{-3+4\sqrt{2}}}) &&
\Q(\sqrt{2},\sqrt{281}) && 2^{12}\cdot 281^4 && 47.413 & \cr
\hlf\vspb
& K_+(\sqrt{\smash{-(35+8\sqrt{17})}\vphantom{-3+4\sqrt{2}}}) &&
\Q(\sqrt{17},\sqrt{137}) && 17^4\cdot 137^4 && 48.259 & \cr
\hlf\vspb
& K_+(\sqrt{\smash{-(29+3\sqrt{13})/2}\vphantom{-3+4\sqrt{2}}}) &&
\Q(\sqrt{13},\sqrt{181}) && 13^4\cdot 181^4 && 48.507 & \cr
\hlf\vspb
& K_+(\sqrt{\smash{-(103+12\sqrt{73})}\vphantom{-3+4\sqrt{2}}}) &&
\Q(\sqrt{73},\sqrt{97}) && 73^4\cdot 97^4 && 84.148 & \cr
\hlf
\endtable
$$

\noindent
Each of these fields is the narrow Hilbert class field of
its quadratic subfield of conductor $f$,
where $f$ is the conductor of $K_+$.
That is, these fields are the narrow Hilbert class fields of
$\Q(\sqrt{34}), \Q(\sqrt{205}), \Q(\sqrt{221}), \Q(\sqrt{305}), \Q(\sqrt{377}),
\Q(\sqrt{545})$,
$\Q(\sqrt{146}), \Q(\sqrt{178}), \Q(\sqrt{745}), \Q(\sqrt{1537}), \Q(\sqrt{466}),
\Q(\sqrt{1945}), \Q(\sqrt{2041}), \Q(\sqrt{562})$,
\linebreak
$\Q(\sqrt{2329}), \Q(\sqrt{2353}),$
and $\Q(\sqrt{7081})$, respectively.
At the same time, each $K$ is the narrow Hilbert class field of $K_+$.

There exist exactly $2$ other dihedral octic CM-fields
with relative class number one.

\def\tableAS{
\vbox\bgroup\offinterlineskip
\halign\bgroup
 \vrule##&\ $##$ \ \hfil &
 \vrule##&\ $##$ \ \hfil &
 \vrule##&\strut\ $##$ \hfil &
 \vrule##&\ \hfil $##$ \ &
 \vrule## \cr                            % preamble
\hlf                                         % full hline
\vspa                                        % wide 2pt
\vspb
& \multispan{1} \hfil {$K$} \hfil &&
\multispan{1} \hfil {$K_+$} \hfil &&
\multispan{1} \hfil {$d_K$}  \hfil &&
rd_K & \cr
\vspb                                        % wide 1pt
\hlf
\vspb
}
$$
\tableAS
& K_+(\sqrt{\smash{-(17+2\sqrt{5})}\vphantom{-3+4\sqrt{2}}}) &&
\Q(\sqrt{5},\sqrt{269}) && 5^4\cdot 269^4 && 36.674 & \cr
\hlf\vspb
& K_+(\sqrt{\smash{-(23+2\sqrt{17})/2}\vphantom{-3+4\sqrt{2}}}) &&
\Q(\sqrt{17},\sqrt{257}) && 17^4\cdot 257^4 && 66.098 & \cr
\hlf
\endtable
$$

\noindent
They are the narrow Hilbert $2$-class fields of
$\Q(\sqrt{1345})$ and $\Q(\sqrt{4573})$, respectively.
At the same time, each $K$ is the narrow Hilbert $2$-class field of $K_+$.
For both of them, $K_+$ has class number three.
\medpagebreak

After this determination, S. Louboutin determined
the CM-fields with class number one
which are the narrow Hilbert class fields of quadratic number fields.
\newpage

(II) $n=12$ ($9$ fields).
\medpagebreak

S. Louboutin, R. Okazaki, and M. Olivier determined
the nonabelian normal CM-fields of degree $12$ with class number one.
There are exactly $9$ such fields and they are all dihedral.
Therefore each of them is the compositum of
a real dihedral sextic field with class number one
and an imaginary quadratic number field with class number one.
(Note that the dihedral group of order twelve is
isomorphic to the direct product of the dihedral group of order six
and the cyclic group of order two: $D_6\cong D_3\times C_2$.)
For each of them, in addition to $d_K$ and $rd_K$,
we give the discriminant $d$ of its cubic subfield and
a generating polynomial $P(X)$ of its cubic subfield,
whose minimal splitting field is its maximal real subfield.

\def\vspaa{height2pt&\omit&&\omit&&\omit&&\omit&&\omit& \cr}
\def\vspbb{height.5pt&\omit&&\omit&&\omit&&\omit&&\omit& \cr}
\def\tableB#1{
\vbox\bgroup\offinterlineskip
\halign\bgroup
 \vrule##&\ $##$ \ \hfil &
 \vrule##&\ \hfil $##$ \ &
 \vrule##&\ $##$ \ \hfil &
 \vrule##&\strut\ $##$ \ \hfil &
 \vrule##&\ \hfil $##$ \ &
 \vrule## \cr                            % preamble
\multispan{8} \hfil {#1} \hfil \cr
\noalign{\vskip 0.3cm}
\hlf                                         % full hline
\vspaa                                       % wide 2pt
\vspbb
& \multispan{1} \hfil {$K$} \hfil && d &&
\multispan{1} \hfil {$P(X)$} \hfil &&
\multispan{1} \hfil {$d_K$} \hfil && rd_K & \cr
\vspbb                                       % wide 1pt
\hlf
\vspbb
}
$$
\tableB{$G=D_6$ ($9$ fields).}
& K_+(\sqrt{-3}) &&  756 && X^3-6X-2 &&
2^8\cdot 3^{14}\cdot 7^6 && 15.131 & \cr
\hlf\vspb
& K_+(\sqrt{-1}) &&  148 && X^3+X^2-3X-1 &&
2^{16}\cdot 37^6 && 15.327 & \cr
\hlf\vspb
& K_+(\sqrt{-3}) && 1620 && X^3-12X-14 &&
2^8\cdot 3^{18}\cdot 5^6 && 18.443 & \cr
\hlf\vspb
& K_+(\sqrt{-3}) && 2700 && X^3-15X-20 &&
2^{12}\cdot 3^{14}\cdot 5^8 && 21.069 & \cr
\hlf\vspb
& K_+(\sqrt{-3}) && 1944 && X^3-9X-6 &&
2^{18}\cdot 3^{22} && 21.196 & \cr
\hlf\vspb
& K_+(\sqrt{-7}) &&  469 && X^3+X^2-5X-4 &&
7^6\cdot 67^6 && 21.656 & \cr
\hlf\vspb
& K_+(\sqrt{-11}) &&  473 && X^3-5X-1 &&
11^6\cdot 43^6 && 21.748 & \cr
\hlf\vspb
& K_+(\sqrt{-3}) && 1425 && X^3-X^2-8X-3 &&
3^6\cdot 5^8\cdot 19^6 && 22.075 & \cr
\hlf\vspb
& K_+(\sqrt{-2}) && 1304 && X^3-X^2-11X-1 &&
2^{18}\cdot 163^6 && 36.110 & \cr
\hlf
\endtable
$$

The fields with discriminants $2^{16}\cdot 37^6$ and $2^8\cdot 3^{18}\cdot 5^6$
are the Hilbert class fields of their sextic subfields
containing $\Q(\sqrt{-37})$ and $\Q(\sqrt{-15})$, respectively,
whereas all the other fields are the narrow Hilbert class fields of $K_+$.
The fields with discriminants $7^6\cdot 67^6, 11^6\cdot 43^6$,
and $2^{18}\cdot 163^6$ are the narrow Hilbert class fields of
the real quadratic number fields $\Q(\sqrt{469}), \Q(\sqrt{473})$
and $\Q(\sqrt{326})$, respectively.
These fields are also the Hilbert class fields of
the imaginary biquadratic bicyclic number fields
$\Q(\sqrt{-7},\sqrt{-67}), \Q(\sqrt{-11},\sqrt{-43})$
and $\Q(\sqrt{-2},\sqrt{-163})$, respectively.

There exist exactly $7$ other dihedral CM-fields of degree $12$
with relative class number one.
\def\tableBS{
\vbox\bgroup\offinterlineskip
\halign\bgroup
 \vrule##&\ $##$ \ \hfil &
 \vrule##&\ \hfil $##$ \ &
 \vrule##&\ $##$ \ \hfil &
 \vrule##&\strut\ $##$ \ \hfil &
 \vrule##&\ \hfil $##$ \ &
 \vrule## \cr                            % preamble
\hlf                                         % full hline
\vspaa                                       % wide 2pt
\vspbb
& \multispan{1} \hfil {$K$} \hfil && d &&
\multispan{1} \hfil {$P(X)$} \hfil &&
\multispan{1} \hfil {$d_K$} \hfil && rd_K & \cr
\vspbb                                       % wide 1pt
\hlf
\vspbb
}
$$
\tableBS
& K_+(\sqrt{-3}) && 4860 && X^3-18X-12 &&
2^{12}\cdot 3^{22}\cdot 5^6 && 33.514 & \cr
\hlf\vspb
& K_+(\sqrt{-1}) &&   940 && X^3-7X-4 &&
2^{12}\cdot 5^6\cdot 47^6 && 33.585 & \cr
\hlf\vspb
& K_+(\sqrt{-2}) &&  4312 && X^3+X^2-16X-8 &&
2^{18}\cdot 7^8\cdot 11^6 && 34.327 & \cr
\hlf\vspb
& K_+(\sqrt{-1}) &&  1708 && X^3-X^2-8X-2 &&
2^{12}\cdot 7^6\cdot 61^6 && 41.327 & \cr
\hlf\vspb
& K_+(\sqrt{-7}) &&  8505 && X^3-27X-51 &&
3^{22}\cdot 5^6\cdot 7^6 && 44.385 & \cr
\hlf\vspb
& K_+(\sqrt{-19}) &&  4104 && X^3-18X-16 &&
2^{18}\cdot 3^{14}\cdot 19^6 && 44.418 & \cr
\hlf\vspb
& K_+(\sqrt{-3}) && 14520 && X^3-33X-22 &&
2^{18}\cdot 3^6\cdot 5^6\cdot 11^8 && 54.181 & \cr
\hlf
\endtable
$$
The class numbers of the maximal real subfields of these fields
are $2$, $2$, $3$, $2$, $2$, $2$, and $2$, respectively.
\newpage

(III) $n=16$ ($12$ fields).
\medpagebreak

S. Louboutin and R. Okazaki determined
the nonabelian normal CM-fields of degree $16$ with class number one.
There are exactly $12$ such fields.
$7$ of them are composita of two normal octic CM-fields,
and the other $5$ fields are
$4$ cyclic octic and $1$ quaternionic extension of
their real quadratic subfields unramified at all finite primes.
\medpagebreak

(i) Composita of two normal octic CM-fields ($7$ fields).
\medpagebreak

Let $K$ be a nonabelian normal CM-field of degree $16$ with class number one
that is the compositum of two normal octic CM-fields $N_1$ and $N_2$.
Then one of them is nonabelian, that is, dihedral or quaternionic.
We may and will assume $N_2$ is so.

If $N_2$ is quaternionic,
then $N_2$ must have class number two and $K$ must be its Hilbert class field,
because quaternionic CM-fields have even (relative) class number.
S. Louboutin proved that
the field $\Q(\sqrt{\smash{-(2+\sqrt{2})(3+\sqrt{3})}\vphantom{-3+4\sqrt{2}}})$
is unique quaternionic CM-field with class number two
and its Hilbert class field matches.

$$
\tableA{$G=Q_8\times C_2$ ($1$ field).}
& K_+(\sqrt{-1}) &&
\Q(\sqrt{\smash{(2+\sqrt{2})(3+\sqrt{3})}\vphantom{-3+4\sqrt{2}}}) &&
2^{48}\cdot 3^{12} && 18.236 & \cr
\hlf
\endtable
$$

For this field,
$N_1=\Q(\sqrt{-1},\sqrt{-2},\sqrt{-3})$.
This field is also the narrow Hilbert class field of $K_+$.

There exist no other CM-field with relative class number one of this type.

If neither $N_1$ nor $N_2$ is quaternionic,
then $N_2$ is dihedral and
$N_1$ is dihedral or abelian of type $(2, 2, 2)$ of $(4,2)$.
In this case, among the three (normal) octic subfields,
two are dihedral and one is abelian.
The Galois group $G$ of $K$ is
isomorphic to $D_4\times C_2$, or $D_4\curlywedge C_4$,
according as the abelian octic subfield is elementary, or not.
\vskip -5mm
$$
\tableA{$G=D_4\times C_2$ ($4$ fields).}
& K_+(\sqrt{-1}) &&
\Q(\sqrt{11},\sqrt{15+8\sqrt{3}}) &&
2^{16}\cdot 3^{12}\cdot 11^8 && 15.120 & \cr
\hlf\vspb
& K_+(\sqrt{-1}) &&
\Q(\sqrt{2},\sqrt{3+\sqrt{3}}) &&
2^{44}\cdot 3^{12} && 15.334 & \cr
\hlf\vspb
& K_+(\sqrt{-3}) &&
\Q(\sqrt{2},\sqrt{\smash{3(5+\sqrt{17})/2}\vphantom{-3+4\sqrt{2}}}) &&
2^{24}\cdot 3^8\cdot 17^8 && 20.199 & \cr
\hlf\vspb
& K_+(\sqrt{\smash{-(5+\sqrt{17})/2}\vphantom{-3+4\sqrt{2}}}) &&
\Q(\sqrt{2},\sqrt{5},\sqrt{17}) &&
2^{24}\cdot 5^8\cdot 17^8 && 26.076 & \cr
\hlf
\endtable
$$

For these fields, $N_1$ and $N_2$ are as follows.

\def\vspac{height2pt&\omit&&\omit& \cr}
\def\vspbc{height.5pt&\omit&&\omit& \cr}
\def\tableC{
\vbox\bgroup\offinterlineskip
\halign\bgroup
 \vrule##&\ $##$ \ \hfil &
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 \vrule## \cr                            % preamble
\hlf                                         % full hline
\vspac                                        % wide 2pt
\vspbc
& \multispan{1} \hfil {$N_1$} \hfil &&
\multispan{1} \hfil {$N_2$} \hfil & \cr
\vspbc                                        % wide 1pt
\hlf
\vspbc
}
$$
\tableC
& \Q(\sqrt{-1},\sqrt{-3},\sqrt{-11}) &&
\Q(\sqrt{11},\sqrt{\smash{-(15+8\sqrt{3})}\vphantom{-3+4\sqrt{2}}}) & \cr
\hlf\vspbc
& \Q(\sqrt{-1},\sqrt{-2},\sqrt{-3}) &&
\Q(\sqrt{2},\sqrt{\smash{-(3+\sqrt{3})}\vphantom{-3+4\sqrt{2}}}) & \cr
\hlf\vspbc
& \Q(\sqrt{-3},\sqrt{2},\sqrt{17}) &&
\Q(\sqrt{2},\sqrt{\smash{-(5+\sqrt{17})/2}\vphantom{-3+4\sqrt{2}}}) & \cr
\hlf\vspbc
& \Q(\sqrt{2},\sqrt{\smash{-(5+\sqrt{17})/2}\vphantom{-3+4\sqrt{2}}}) &&
\Q(\sqrt{2},\sqrt{\smash{-5(5+\sqrt{17})}\vphantom{-3+4\sqrt{2}}}) & \cr
\hlf
\endtable
$$

These fields are also the Hilbert class fields of
$N_2, N_2, N_1$ and $N_2$, respectively.
The third and fourth fields are dihedral extensions of
the real quadratic number fields
$F=\Q(\sqrt{102})$ and $\Q(\sqrt{170})$, respectively,
that are unramified at all finite primes.

There exist no other CM-field with relative class number one of this type.
\vskip -5mm
$$
\tableA{$G=D_4\curlywedge C_4$ ($2$ fields).}
& K_+(\sqrt{\smash{-(5+\sqrt{17})/2}\vphantom{-3+4\sqrt{2}}}) &&
\Q(\sqrt{2},\sqrt{17+4\sqrt{17}}) &&
2^{24}\cdot 17^{12} && 23.679 & \cr
\hlf\vspb
& K_+(\sqrt{\smash{-(13+2\sqrt{13})}\vphantom{-3+4\sqrt{2}}}) &&
\Q(\sqrt{17},\sqrt{\smash{(143+31\sqrt{13})/2}\vphantom{-3+4\sqrt{2}}}) &&
13^{12}\cdot 17^8 && 28.228 & \cr
\hlf
\endtable
$$

For these fields, $N_1$ and $N_2$ are as follows.

$$
\tableC
& \Q(\sqrt{2},\sqrt{\smash{-(5+\sqrt{17})/2}\vphantom{-3+4\sqrt{2}}}) &&
\Q(\sqrt{2},\sqrt{\smash{-(17+3\sqrt{17})/2}\vphantom{-3+4\sqrt{2}}}) & \cr
\hlf\vspbc
& \Q(\sqrt{17},\sqrt{\smash{-(13+2\sqrt{13})}\vphantom{-3+4\sqrt{2}}}) &&
\Q(\sqrt{17},\sqrt{\smash{-(9+\sqrt{13})/2}\vphantom{-3+4\sqrt{2}}}) & \cr
\hlf
\endtable
$$

For the former field, $N_1$ are dihedral,
but for the latter field, $N_1$ is abelian of type $(2,4)$.
These fields are also the Hilbert class fields of
$N_2$ and $N_1$, respectively.

There exist no other CM-field with relative class number one of this type.

\medpagebreak

(ii) Unramified non-dihedral extensions of real quadratic number fields
($5$ fields).
\def\vspaaa{height2pt&\omit&&\omit&&\omit& \cr}
\def\vspbbb{height.5pt&\omit&&\omit&&\omit& \cr}
$$
\vbox{
\offinterlineskip
% preamble
\halign{
\vrule#&
\ $#$ \ \hfil & \vrule# & \ $#$\strut \ \hfil & \vrule# & \ \hfil $#$ \ &
\vrule# \cr
\multispan{6} \hfil {$G=D_8$ ($4$ fields).} \hfil \cr
\noalign{\vskip 0.3cm}
\vspbbb
\noalign{\hrule}
\vspaaa
& \multispan{1} \hfil {$K_+$} \hfil &&
\multispan{1} \hfil {$d_K$}  \hfil &&
rd_K & \cr
\vspbbb
\noalign{\hrule}
\vspbbb
& \Q(\sqrt{101},\sqrt{11+2\sqrt{5}}) &&
5^8\cdot 101^8 && 22.472 & \cr
\hlf\vspbbb
& \Q(\sqrt{53},\sqrt{51+14\sqrt{13}}) &&
13^8\cdot 53^8 && 26.248 & \cr
\hlf\vspbbb
& \Q(\sqrt{61},\sqrt{23+6\sqrt{13}}) &&
13^8\cdot 61^8 && 28.160 & \cr
\hlf\vspbbb
& \Q(\sqrt{181},\sqrt{19+6\sqrt{5}}) &&
5^8\cdot 181^8 && 30.083 & \cr
\noalign{\hrule}
}}
$$

These fields are
the narrow Hilbert class fields of the real quadratic number fields
$\Q(\sqrt{505}), \Q(\sqrt{689}), \Q(\sqrt{793})$ and $\Q(\sqrt{905})$.

There exists only $1$ other dihedral CM-field of degree $16$
with relative class number one.
$$
\vbox{
\offinterlineskip
% preamble
\halign{
\vrule#&
\ $#$ \ \hfil & \vrule# & \ $#$\strut \ \hfil & \vrule# & \ \hfil $#$ \ &
\vrule# \cr
\vspbbb
\noalign{\hrule}
\vspaaa
& \multispan{1} \hfil {$K_+$} \hfil &&
\multispan{1} \hfil {$d_K$}  \hfil &&
rd_K & \cr
\vspbbb
\noalign{\hrule}
\vspbbb
& \Q(\sqrt{257},\sqrt{17+4\sqrt{2}}) &&
2^{24}\cdot 257^8 && 45.343 & \cr
\noalign{\hrule}
}}
$$

This field is
the narrow Hilbert class fields of the real quadratic number field
$\Q(\sqrt{514})$.
For this $K$, $K_+$ has class number three.

%These $K$ can be written in the form $K_+(\sqrt{\alpha})$,
%where $\alpha$ is an integer of a nonnormal quartic subfield of $K$
%over which $K$ is biquadratic bicyclic.
%However, we do not give explicit forms of $\alpha$.
\vskip -5mm
$$
\tableA{$G=D_4\curlyvee C_4$ ($1$ field).}
& K_+(\sqrt{\smash{-(2\sqrt{2}+3\sqrt{5})(2+\sqrt{5})}
\vphantom{-3+4\sqrt{2}}}) &&
\Q(\sqrt{2},\sqrt{5},\sqrt{37}) &&
2^{24}\cdot 5^8\cdot 37^8 && 38.470 & \cr
\hlf
\endtable
$$
This field is the unique quaternionic extension of
the real quadratic number field $\Q(\sqrt{370})$
unramified at all finite primes which is normal over $\Q$.
This is also the common narrow Hilbert class field of
its biquadratic subfields and also that of $K_+$.
%$\Q(\sqrt{2},\sqrt{185}), \Q(\sqrt{5},\sqrt{74})$
%and $\Q(\sqrt{37},\sqrt{10})$.
%Moreover, this field is also the narrow Hilbert class field of $K_+$.

There exist no other CM-field with relative class number one of this type.
\newpage

S. Louboutin, R. Okazaki, and M. Olivier proved that
a nonabelian normal CM-field of degree $<32$ must have degree
$8, 12, 16, 20, 24,$ or $28$,
and that if it has degree $20$ or $28$, then it must be dihedral.
Y. Lefeuvre determined the dihedral CM-fields with class number one.
There exist only two such fields of degrees $>16$
and their degrees are $20$ and $24$.
\medpagebreak

(IV) $n=20$ ($1$ field).
\medpagebreak

There is exactly $1$ CM-field of degree $20$ and this is dihedral.
Therefore this is the compositum of
a real dihedral field of degree $10$ with class number one
and an imaginary quadratic number field with class number one.
For this, in addition to $d_K$ and $rd_K$,
we give the discriminant $d$ of its quintic subfield and
a generating polynomial $P(X)$ of its quintic subfield,
whose minimal splitting field is its maximal real subfield.

$$
\tableB{$G=D_{10}$ ($1$ field).}
& K_+(\sqrt{-19}) && 667489 && X^5-X^4-6X^3+5X^2+3X-1 &&
19^{10}\cdot 43^{10} && 28.583 & \cr
\hlf
\endtable
$$
This field is the narrow Hilbert class fields of
the real quadratic number field $\Q(\sqrt{817})$.

There exists exactly $1$ other dihedral CM-field of degree $20$
with relative class number one.
\vskip -5mm
$$
\tableBS
& K_+(\sqrt{-2}) && 49000000 && X^5-35X^3-30X^2+10X+4 &&
2^{30}\cdot 5^{26}\cdot 7^{10} && 30.190 & \cr
\hlf
\endtable
$$
For this $K$, $K_+$ has class number two.
\medpagebreak

(V) $n=24$ ($7$ fields).
\medpagebreak

(i) Dihedral ($1$ field).

$$
\tableA{$G=D_{12}$ ($1$ field).}
& K_+(\sqrt{\smash{-(17+2\sqrt{5})}\vphantom{-3+4\sqrt{2}}}) &&
F(\sqrt{5},a_{31}) && 5^{12}\cdot 269^{12} && 36.674 & \cr
\hlf
\endtable
$$
Here, $a_{31}$ denotes any algebraic integer
generating $31$th totally real nonnormal cubic number field.
For example, we may take it as a root of the cubic polynomial $X^3-7X-1$.
This field is the narrow Hilbert class fields of
the real quadratic number field $F=\Q(\sqrt{1345})$.
There exist no other dihedral CM-field of degree $24$
with relative class number one.
\medpagebreak

(ii) $\Gal(K_+/\Q)\cong A_4$ ($3$ fields).
\medpagebreak

F. Lemmermeyer, S. Louboutin, and R. Okazaki determined
the normal CM-fields with class number one
whose maximal totally real subfields have Galois group
isomorphic to $A_4$.
There are exactly $3$ such fields:
$2$ of them are $A_4\times C_2$-extensions of $\Q$ and
the other is an $\SL(2,3)$-extension of $\Q$.
For each of them, in addition to $d_K$ and $rd_K$,
we give the conductor of the unique (cyclic) cubic subfield
and a generating polynomial $P(X)$ of
a nonnormal sextic or octic CM-subfield whose normal closure is $K$.

\def\tableD#1{
\vbox\bgroup\offinterlineskip
\halign\bgroup
 \vrule##&\ $##$ \ \hfil &
 \vrule##&\ \hfil $##$ \ &
 \vrule##&\ $##$ \ \hfil &
 \vrule##&\strut\ $##$ \ \hfil &
 \vrule##&\ \hfil $##$ \ &
 \vrule## \cr                            % preamble
\multispan{8} \hfil {#1} \hfil \cr
\noalign{\vskip 0.3cm}
\hlf                                         % full hline
\vspaa                                       % wide 2pt
\vspbb
& \multispan{1} \hfil {$K$} \hfil && f &&
\multispan{1} \hfil {$P(X)$} \hfil &&
\multispan{1} \hfil {$d_K$} \hfil && rd_K & \cr
\vspbb                                       % wide 1pt
\hlf
\vspbb
}
$$
\tableD{$G=A_4\times C_2$ ($2$ fields).}
& K_+(\sqrt{-2}) &&  31 && X^6+52X^4+457X^2+2 &&
2^{36}\cdot 31^{16} && 27.911 & \cr
\hlf\vspb
& K_+(\sqrt{-3}) &&  61 && X^6+121X^4+3250X^2+3 &&
3^{12}\cdot 61^{16} && 36.839 & \cr
\hlf
\endtable
$$

This field is
the Hilbert class field of the imaginary cyclic sextic number field
of conductor 248 whose quadratic subfield is $\Q(\sqrt{-2})$.
The field with discriminant $3^{12}\cdot 61^{16}$ is
the Hilbert class field of the imaginary cyclic sextic number field
of conductor 201 whose quadratic subfield is $\Q(\sqrt{-3})$.

There exist no other dihedral CM-field of degree $24$
with relative class number one.

$$
\tableD{$G=\SL(2,3)$ ($1$ field).}
&  &&  163 && X^8+9X^6+23X^4+14X^2+1 &&
163^{16} && 29.839 & \cr
\hlf
\endtable
$$

This field is
the unique quaternionic extension of
the cyclic cubic number field of conductor $163$
unramified at all finite primes.

There exist no other dihedral CM-field of degree $24$
with relative class number one.
\medpagebreak

(iii) $G\cong D_{m}\times C_{m'}$ ($mm'=12$) ($3$ fields).
\medpagebreak

Y.-H. Park determined
the normal CM-fields with class number one
whose Galois groups are isomorphic to one of $5$ groups
$Q_{12}\times C_2$, $D_4\ltimes C_3$, $D_6\times C_2$, $D_4\times C_3$,
and $D_3\times C_4$.
For $Q_{12}\times C_2$ and $D_4\ltimes C_3$,
there is no such field,
and for each of the other $3$ groups,
there is exactly $1$ such field.
All of them are composita of a nonnormal totally real cubic field
defined by $f(X)$ and a normal octic CM-field $N$.
For each of them, in addition to $d_K$ and $rd_K$,
we give a defining polynomial $f(X)$ of
a nonnormal totally real cubic field and a normal octic CM-subfield $N$.

\def\tableE#1{
\vbox\bgroup\offinterlineskip
\halign\bgroup
 \vrule##&\ $##$ \ \hfil &
 \vrule##&\ $##$ \ \hfil &
 \vrule##&\strut\ $##$ \hfil &
 \vrule##&\ \hfil $##$ \ &
 \vrule## \cr                            % preamble
\multispan{8} \hfil {#1} \hfil \cr
\noalign{\vskip 0.3cm}
\hlf                                         % full hline
\vspa                                        % wide 2pt
\vspb
& \multispan{1} \hfil {$N$} \hfil &&
\multispan{1} \hfil {$f(X)$} \hfil &&
\multispan{1} \hfil {$d_K$}  \hfil &&
rd_K & \cr
\vspb                                        % wide 1pt
\hlf
\vspb
}
$$
\tableE{$G=D_6\times C_2$ ($1$ field).}
& \Q(\sqrt{-3},\sqrt{-1},\sqrt{-7}) &&
X^3-6X-2 && 2^{32}\cdot 3^{28}\cdot 7^{12} && 24.019 & \cr
\hlf
\endtable
$$
$$
\tableE{$G=D_4\times C_3$ ($1$ field).}
& \Q(\sqrt{17},\sqrt{\smash{-(9+\sqrt{13})/2}\vphantom{-3+4\sqrt{2}}})
&& X^3-X^2-4X-1 && 13^{20}\cdot 17^{12} && 89.527 & \cr
\hlf
\endtable
$$
$$
\tableE{$G=D_3\times C_4$ ($1$ field).}
& \Q(\zeta_5,\sqrt{13}) &&
X^3-10X-10 && 2^{16}\cdot 5^{22}\cdot 13^{12} && 59.024 & \cr
\hlf
\endtable
$$

There exist no other CM-fields
with relative class number one of these types.
\newpage

(VI) $n=32$ (at least $4$ fields).
\medpagebreak

(i) Composita of two normal CM-fields of degree $16$ ($4$ fields).
\medpagebreak

Let $K$ be a nonabelian normal CM-field of degree $32$ with class number one
that is the compositum of two normal octic CM-fields $M_1$ and $M_2$.

$$
\tableA{$G=(Q_8\curlyvee C_4)\times C_2$ ($1$ field).}
& K_+(\sqrt{-1}) &&
\Q(\sqrt{\smash{2+\sqrt{2}}\vphantom{-3+4\sqrt{2}}},
\sqrt{\smash{3+\sqrt{3}}\vphantom{-3+4\sqrt{2}}}) &&
2^{?}\cdot 3^{?} && ? & \cr
\hlf
\endtable
$$

(VII) $n=36$ ($12$ fields).
\medpagebreak

K.-Y. Chang and S.-H. Kwon determined
the nonabelian normal CM-fields of degree $36$ with class number one.
There are exactly three such fields and
all of them have Galois group isomorphic to
$D_6\times C_3\cong D_3\times C_3 \times C_2$.
Therefore each $K$ of them is
the compositum of $K_+$,
that is a real $D_3\times C_3$-extension of $\Q$ with class number one
and an imaginary quadratic number field with class number one,
and moreover $K_+$ is the compositum of
a real sextic dihedral field $L$ and a real cycic cubic field $F$.
For each of them, in addition to $d_K$ and $rd_K$,
we give the discriminant $d$ of any nonnormal cubic subfield of $L$ and
its generating polynomial $P(X)$,
and the conductor $f$ of $F$.

\def\vspc{height2pt&\omit&&\omit&&\omit&&\omit&&\omit&&\omit& \cr}
\def\vspd{height.5pt&\omit&&\omit&&\omit&&\omit&&\omit&&\omit& \cr}
\def\tableF#1{
\vbox\bgroup\offinterlineskip
\halign\bgroup
 \vrule##&\ $##$ \ \hfil &
 \vrule##&\ \hfil $##$ \ &
 \vrule##&\ $##$ \ \hfil &
 \vrule##&\ \hfil $##$ \ &
 \vrule##&\strut\ $##$ \ \hfil &
 \vrule##&\ \hfil $##$ \ &
 \vrule## \cr                            % preamble
\multispan{10} \hfil {#1} \hfil \cr
\noalign{\vskip 0.3cm}
\hlf                                         % full hline
\vspc                                      % wide 2pt
\vspd
& \multispan{1} \hfil {$K$} \hfil && d &&
\multispan{1} \hfil {$P(X)$} \hfil &&
\multispan{1} \hfil {$f$} \hfil &&
\multispan{1} \hfil {$d_K$} \hfil && rd_K & \cr
\vspd                                      % wide 1pt
\hlf
\vspd
}
$$
\tableF{$G=D_6\times C_3$ ($3$ fields).}
& K_+(\sqrt{-3}) &&  756 && X^3-6X-2 && 9 &&
2^{24}\cdot 3^{62}\cdot 7^{18} && 27.857 & \cr
\hlf\vspd
& K_+(\sqrt{-2}) && 4312 && X^3+X^2-16X-8 && 7 &&
2^{54}\cdot 7^{24}\cdot 11^{18} && 34.327 & \cr
\hlf\vspd
& K_+(\sqrt{-3}) && 1620 && X^3-12X-14 && 9 &&
2^{24}\cdot 3^{66}\cdot 5^{18} && 48.249 & \cr
\hlf
\endtable
$$

The field with discriminant $2^{54}\cdot 7^{24}\cdot 11^{18}$
is the narrow Hilbert class field of the splitting field of $P(X)$
and at the same time, this field is
the Hilbert class field of its dihedral CM-subfield of degree $12$.

There exists no other CM-field of degree $36$
with relative class number one.
\medpagebreak

(VIII) $n=40$ ($1$ field).
\medpagebreak

Y.-H. Park determined
the nonabelian normal CM-fields of degree $40$ with class number one.
There is exactly one such field
and this has Galois group isomorphic to $F_{20}\times C_2$.
Therefore this field is the compositum of
a $F_{20}$-extension of $\Q$
and a quadratic number field.
For each of them, in addition to $d_K$ and $rd_K$,
we give the discriminant $d$ of its quintic subfield and
a generating polynomial $P(X)$ of its quintic subfield,
whose minimal splitting field is its maximal real subfield.

$$
\tableB{$G=F_{20}\times C_2$ ($1$ field).}
& K_+(\sqrt{-7}) && 2450000 && X^5-10X^3+20X+10 &&
2^{32}\cdot 5^{46}\cdot 7^{20} && 29.321 & \cr
\hlf
\endtable
$$

There exists no other CM-field of degree $40$
with relative class number one.
\enddocument
