LIST OF KNOWN NONABELIAN NORMAL

CM-FIELDS WITH CLASS NUMBER ONE

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 CMfields 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}.)

\begin{matrix}n & G & #_{1} & #_{2} & # \\ 8 & D_{4} & 17 & 2 & 19 \\ 12 & D_{6} & 9 & 7 & 16 \\ 16 & Q_{8} × C_{2} & 1 & 0 & 1 \\ 16 & D_{4} × C_{2} & 4 & 0 & 4 \\ 16 & D_{4} ⋏ C_{4} & 2 & 0 & 2 \\ 16 & D_{8} & 4 & 1 & 5 \\ 16 & D_{4} ⋎ C_{4} & 1 & 0 & 1 \\ 20 & D_{10} & 1 & 1 & 2 \\ 24 & D_{12} & 1 & 0 & 1 \\ 24 & A_{4} × C_{2} & 2 & 0 & 2 \\ 24 & SL(2, 3) & 1 & 0 & 1 \\ 24 & D_{6} × C_{2} & 1 & 0 & 1 \\ 24 & D_{3} × C_{4} & 1 & 0 & 1 \\ 24 & D_{4} × C_{3} & 1 & 0 & 1 \\ 32 & (Q_{8} ⋏ C_{4}) × C_{2} & 1 & 0 & 1 \\ 32 & 32Γ_{2}d & 1 & 0 & 1 \\ 32 & 32Γ_{2}j_{1} & 1 & 0 & 1 \\ 32 & M_{16} × C_{2} & 3 & 0 & 3 \\ 36 & D_{6} × C_{3} & 3 & 0 & 3 \\ 40 & F_{20} × C_{2} & 1 & 0 & 1 \\ 48 & D_{8} ⋊ C_{3} & 1 & 0 & 1 \end{matrix}

Typeset by AMS-TEX

As usual, C_{n} is the cyclic group of order n, D_{n} (n ≧ 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).

(I) n = 8 (17 fields).

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.

G = D_{4} (17 fields).
K  K_{+}  d_{K}  rd_{K}
K_{+}(\sqrt{−(5 + 2\sqrt{2})})  Q(\sqrt{2}, \sqrt{17})  2^{12} · 17^{4} 11.661
K_{+}(\sqrt{−(13 + \sqrt{5})/2})  Q(\sqrt{5}, \sqrt{41})  5^{4} · 41^{4} 14.317
K_{+}(\sqrt{−(9 + \sqrt{13})/2})  Q(\sqrt{13}, \sqrt{17}) 13^{4} · 17^{4} 14.866
K_{+}(\sqrt{−(17 + 3\sqrt{5})/2}) Q(\sqrt{5}, \sqrt{61})  5^{4} · 61^{4} 17.464
K_{+}(\sqrt{−(9 + 2\sqrt{13})})  Q(\sqrt{13}, \sqrt{29}) 13^{4} · 29^{4} 19.416
K_{+}(\sqrt{−(21 + \sqrt{5})/2})  Q(\sqrt{5}, \sqrt{109}) 5^{4} · 109^{4} 23.345
K_{+}(\sqrt{−(9 + 2\sqrt{2})})  Q(\sqrt{2}, \sqrt{73})  2^{12} · 73^{4} 24.166
K_{+}(\sqrt{−(11 + 4\sqrt{2})})  Q(\sqrt{2}, \sqrt{89})  2^{12} · 89^{4} 26.683
K_{+}(\sqrt{−(13 + 2\sqrt{5})})  Q(\sqrt{5}, \sqrt{149}) 5^{4} · 149^{4} 27.294
K_{+}(\sqrt{−(13 + 2\sqrt{29})})  Q(\sqrt{29}, \sqrt{53}) 29^{4} · 53^{4} 39.204
K_{+}(\sqrt{−(19 + \sqrt{233})/2}) Q(\sqrt{2}, \sqrt{233}) 2^{12} · 233^{4} 43.174
K_{+}(\sqrt{−(41 + 5\sqrt{5})/2}) Q(\sqrt{5}, \sqrt{389}) 5^{4} · 389^{4} 44.102
K_{+}(\sqrt{−(25 + 6\sqrt{13})})  Q(\sqrt{13}, \sqrt{157}) 13^{4} · 157^{4} 45.177
K_{+}(\sqrt{−(17 + 2\sqrt{2})})  Q(\sqrt{2}, \sqrt{281}) 2^{12} · 281^{4} 47.413
K_{+}(\sqrt{−(35 + 8\sqrt{17})})  Q(\sqrt{17}, \sqrt{137}) 17^{4} · 137^{4} 48.259
K_{+}(\sqrt{−(29 + 3\sqrt{13})/2}) Q(\sqrt{13}, \sqrt{181}) 13^{4} · 181^{4} 48.507
K_{+}(\sqrt{−(103 + 12\sqrt{73})}) Q(\sqrt{73}, \sqrt{97}) 73^{4} · 97^{4} 84.148

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}), 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.

\begin{matrix}K & K_{+} & d_{K} & rd_{K} \\ K_{+}(\sqrt{−(17 + 2\sqrt{5})}) & Q(\sqrt{5}, \sqrt{269}) & 5^{4} · 269^{4} & 36.674 \\ K_{+}(\sqrt{−(23 + 2\sqrt{17})/2}) & Q(\sqrt{17}, \sqrt{257}) & 17^{4} · 257^{4} & 66.098 \end{matrix}

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.

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

(II) n = 12 (9 fields).

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} ≅ D_{3} × 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.

G = D_{6} (9 fields).
K  d  P (X)  d_{K}  rd_{K}
K_{+}(\sqrt{−3})  756 X^{3} − 6X − 2  2^{8} · 3^{14} · 7^{6} 15.131
K_{+}(\sqrt{−1})  148 X^{3} + X^{2} − 3X − 1 2^{16} · 37^{6}  15.327
K_{+}(\sqrt{−3}) 1620 X^{3} − 12X − 14  2^{8} · 3^{18} · 5^{6} 18.443
K_{+}(\sqrt{−3}) 2700 X^{3} − 15X − 20  2^{12} · 3^{14} · 5^{8} 21.069
K_{+}(\sqrt{−3}) 1944 X^{3} − 9X − 6  2^{18} · 3^{22}  21.196
K_{+}(\sqrt{−7})  469 X^{3} + X^{2} − 5X − 4 7^{6} · 67^{6}  21.656
K_{+}(\sqrt{−11}) 473 X^{3} − 5X − 1  11^{6} · 43^{6}  21.748
K_{+}(\sqrt{−3}) 1425 X^{3} − X^{2} − 8X − 3 3^{6} · 5^{8} · 19^{6} 22.075
K_{+}(\sqrt{−2}) 1304 X^{3} − X^{2} − 11X − 1 2^{18} · 163^{6}  36.110

The fields with discriminants 2^{16} · 37^{6} and 2^{8} · 3^{18} · 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} ·67^{6}, 11^{6} ·43^{6}, and 2^{18} ·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.

\begin{matrix}K & d & P (X) & d_{K} & rd_{K} \\ K_{+}(\sqrt{−3}) & 4860 & X^{3} − 18X − 12 & 2^{12} · 3^{22} · 5^{6} & 33.514 \\ K_{+}(\sqrt{−1}) & 940 & X^{3} − 7X − 4 & 2^{12} · 5^{6} · 47^{6} & 33.585 \\ K_{+}(\sqrt{−2}) & 4312 & X^{3} + X^{2} − 16X − 8 & 2^{18} · 7^{8} · 11^{6} & 34.327 \\ K_{+}(\sqrt{−1}) & 1708 & X^{3} − X^{2} − 8X − 2 & 2^{12} · 7^{6} · 61^{6} & 41.327 \\ K_{+}(\sqrt{−7}) & 8505 & X^{3} − 27X − 51 & 3^{22} · 5^{6} · 7^{6} & 44.385 \\ K_{+}(\sqrt{−19}) & 4104 & X^{3} − 18X − 16 & 2^{18} · 3^{14} · 19^{6} & 44.418 \\ K_{+}(\sqrt{−3}) & 14520 & X^{3} − 33X − 22 & 2^{18} · 3^{6} · 5^{6} · 11^{8} & 54.181 \end{matrix}

The class numbers of the maximal real subfields of these fields are 2, 2, 3, 2, 2, 2, and 2, respectively.

(III) n = 16 (12 fields).

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.

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

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{−(2 + \sqrt{2})(3 + \sqrt{3})}) is unique quaternionic CM-field with class number two and its Hilbert class field matches.

G = Q_{8} × C_{2} (1 field).
K  K_{+}  d_{K}  rd_{K}
K_{+}(\sqrt{−1}) Q(\sqrt{(2 + \sqrt{2})(3 + \sqrt{3})}) 2^{48} · 3^{12} 18.236

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} × C_{2}, or D_{4} ⋏ C_{4}, according as the abelian octic subfield is elementary, or not.

G = D_{4} × C_{2} (4 fields).
K  K_{+}  d_{K}  rd_{K}
K_{+}(\sqrt{−1})  Q(\sqrt{11}, \sqrt{15 + 8\sqrt{3}})  2^{16} · 3^{12} · 11^{8} 15.120
K_{+}(\sqrt{−1})  Q(\sqrt{2}, \sqrt{3 + \sqrt{3}})  2^{44} · 3^{12}  15.334
K_{+}(\sqrt{−3})  Q(\sqrt{2}, \sqrt{3(5 + \sqrt{17})/2}) 2^{24} · 3^{8} · 17^{8} 20.199
K_{+}(\sqrt{−(5 + \sqrt{17})/2}) Q(\sqrt{2}, \sqrt{5}, \sqrt{17})  2^{24} · 5^{8} · 17^{8} 26.076

For these fields, N_{1} and N_{2} are as follows.

N_{1}  N_{2}
Q(\sqrt{−1}, \sqrt{−3}, \sqrt{−11})  Q(\sqrt{11}, \sqrt{−(15 + 8\sqrt{3})})
Q(\sqrt{−1}, \sqrt{−2}, \sqrt{−3})  Q(\sqrt{2}, \sqrt{−(3 + \sqrt{3})})
Q(\sqrt{−3}, \sqrt{2}, \sqrt{17})  Q(\sqrt{2}, \sqrt{−(5 + \sqrt{17})/2})
Q(\sqrt{2}, \sqrt{−(5 + \sqrt{17})/2}) Q(\sqrt{2}, \sqrt{−5(5 + \sqrt{17})})

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.

G = D_{4} ⋏ C_{4} (2 fields).
K  K_{+}  d_{K}  rd_{K}
K_{+}(\sqrt{−(5 + \sqrt{17})/2}) Q(\sqrt{2}, \sqrt{17 + 4\sqrt{17}})  2^{24} · 17^{12} 23.679
K_{+}(\sqrt{−(13 + 2\sqrt{13})}) Q(\sqrt{17}, \sqrt{(143 + 31\sqrt{13})/2}) 13^{12} · 17^{8} 28.228

For these fields, N_{1} and N_{2} are as follows.

\begin{matrix}N_{1} & N_{2} \\ Q(\sqrt{2}, \sqrt{−(5 + \sqrt{17})/2}) & Q(\sqrt{2}, \sqrt{−(17 + 3\sqrt{17})/2}) \\ Q(\sqrt{17}, \sqrt{−(13 + 2\sqrt{13})}) & Q(\sqrt{17}, \sqrt{−(9 + \sqrt{13})/2}) \end{matrix}

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.

(ii) Unramified non-dihedral extensions of real quadratic number fields (5 fields).

G = D_{8} (4 fields).
K_{+}  d_{K}  rd_{K}
Q(\sqrt{101}, \sqrt{11 + 2\sqrt{5}}) 5^{8} · 101^{8} 22.472
Q(\sqrt{53}, \sqrt{51 + 14\sqrt{13}}) 13^{8} · 53^{8} 26.248
Q(\sqrt{61}, \sqrt{23 + 6\sqrt{13}}) 13^{8} · 61^{8} 28.160
Q(\sqrt{181}, \sqrt{19 + 6\sqrt{5}}) 5^{8} · 181^{8} 30.083

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.

\begin{matrix}K_{+} & d_{K} & rd_{K} \\ Q(\sqrt{257}, \sqrt{17 + 4\sqrt{2}}) & 2^{24} · 257^{8} & 45.343 \end{matrix}

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.

G = D_{4} ⋎ C_{4} (1 field).
K  K_{+}  d_{K}  rd_{K}
K_{+}(\sqrt{−(2\sqrt{2} + 3\sqrt{5})(2 + \sqrt{5})}) Q(\sqrt{2}, \sqrt{5}, \sqrt{37}) 2^{24} · 5^{8} · 37^{8} 38.470

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_{+}.

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

S. Louboutin, R. Okazaki, and M. Olivier proved that a nonabelian normal CMfield 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.

(IV) n = 20 (1 field).

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.

G = D_{10} (1 field).
K  d  P (X)  d_{K}  rd_{K}
K_{+}(\sqrt{−19}) 667489 X^{5} − X^{4} − 6X^{3} + 5X^{2} + 3X − 1 19^{10} · 43^{10} 28.583

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.

\begin{matrix}K & d & P (X) & d_{K} & rd_{K} \\ K_{+}(\sqrt{−2}) & 49000000 & X^{5} − 35X^{3} − 30X^{2} + 10X + 4 & 2^{30} · 5^{26} · 7^{10} & 30.190 \end{matrix}

For this K, K_{+} has class number two.

(V) n = 24 (7 fields).

(i) Dihedral (1 field).

G = D_{12} (1 field).
K  K_{+}  d_{K}  rd_{K}
K_{+}(\sqrt{−(17 + 2\sqrt{5})}) F (\sqrt{5}, a_{31}) 5^{12} · 269^{12} 36.674

Here, a_{31} denotes any algebraic integer generating 31th 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.

(ii) Gal(K_{+}/Q) ≅ A_{4} (3 fields).

F. Lemmermeyer, S. Louboutin, and R. Okazaki determined the normal CMfields 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} ×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.

G = A_{4} × C_{2} (2 fields).
K  f  P (X)  d_{K}  rd_{K}
K_{+}(\sqrt{−2}) 31 X^{6} + 52X^{4} + 457X^{2} + 2  2^{36} · 31^{16} 27.911
K_{+}(\sqrt{−3}) 61 X^{6} + 121X^{4} + 3250X^{2} + 3 3^{12} · 61^{16} 36.839

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} · 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.

G = SL(2, 3) (1 field).
K f  P (X)  d_{K}  rd_{K}
163 X^{8} + 9X^{6} + 23X^{4} + 14X^{2} + 1 163^{16} 29.839

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.

(iii) G ≅ D_{m} × C_{m′} (mm′ = 12) (3 fields).

Y.-H. Park determined the normal CM-fields with class number one whose Galois groups are isomorphic to one of 5 groups Q_{12} × C_{2}, D_{4} ⋉ C_{3}, D_{6} × C_{2}, D_{4} × C_{3}, and D_{3} × C_{4}. For Q_{12} × C_{2} and D_{4} ⋉ 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.

G = D_{6} × C_{2} (1 field).
N  f(X)  d_{K}  rd_{K}
Q(\sqrt{−3}, \sqrt{−1}, \sqrt{−7}) X^{3} − 6X − 2 2^{32} · 3^{28} · 7^{12} 24.019
G = D_{4} × C_{3} (1 field).
N  f(X)  d_{K}  rd_{K}
Q(\sqrt{17}, \sqrt{−(9 + \sqrt{13})/2}) X^{3} − X^{2} − 4X − 1 13^{20} · 17^{12} 89.527
G = D_{3} × C_{4} (1 field).
N  f(X)  d_{K}  rd_{K}
Q(ζ_{5}, \sqrt{13}) X^{3} − 10X − 10 2^{16} · 5^{22} · 13^{12} 59.024

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

(VI) n = 32 (at least 4 fields).

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

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}.

G = (Q_{8} ⋎ C_{4}) × C_{2} (1 field).
K  K_{+}  d_{K} rd_{K}
K_{+}(\sqrt{−1}) Q(\sqrt{2 + \sqrt{2}}, \sqrt{3 + \sqrt{3}}) 2^{?} · 3^{?}  ?

(VII) n = 36 (12 fields).

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} × C_{3} ≅ D_{3} × C_{3} × C_{2}. Therefore each K of them is the compositum of K_{+}, that is a real D_{3} × 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.

G = D_{6} × C_{3} (3 fields).
K  d  P (X)  f  d_{K}  rd_{K}
K_{+}(\sqrt{−3}) 756 X^{3} − 6X − 2  9 2^{24} · 3^{62} · 7^{18} 27.857
K_{+}(\sqrt{−2}) 4312 X^{3} + X^{2} − 16X − 8 7 2^{54} · 7^{24} · 11^{18} 34.327
K_{+}(\sqrt{−3}) 1620 X^{3} − 12X − 14  9 2^{24} · 3^{66} · 5^{18} 48.249

The field with discriminant 2^{54} · 7^{24} · 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.

(VIII) n = 40 (1 field).

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} × 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.

G = F_{20} × C_{2} (1 field).
K  d  P (X)  d_{K}  rd_{K}
K_{+}(\sqrt{−7}) 2450000 X^{5} − 10X^{3} + 20X + 10 2^{32} · 5^{46} · 7^{20} 29.321

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