Descending Central Series of

2-Stage Metabelian 3-Groups

1. The classical scenario.

The original purpose of Brigitte Nebelung's thesis [3] , written in 1989 under the supervision of Wolfram Jehne at Cologne, was to gain a thorough overview of all 2-stage metabelian 3-groups G with commutator factor group G/G' of type (3,3) and to apply the results to the following problems which were first posed by Arnold Scholz and Olga Taussky-Todd in 1933 (inspired by the Furtwängler / Artin proof of the principal ideal theorem) for imaginary quadratic fields K [1] but arise for any algebraic number field K with 3-class group Syl₃C(K) of type (3,3):

1. to determine the Galois group G(K₂|K) of the 2^nd Hilbert 3-class field K₂ of K over K,
2. to find the structure of the 3-class group Syl₃C(K₁) of the 1^st Hilbert 3-class field K₁ of K.

				K₂
				\|
				K₁
				\|
				K

G(K₂\|K₂) = 1
\|
G' = G(K₂\|K₁) = Syl₃C(K₁)	----
\|		G/G' = G(K₁\|K) = Syl₃C(K) = (3,3)
G = G(K₂\|K)	----

This application is due to class field theory, since K₁ is the maximal abelian unramified 3-extension of K and thus the subgroup U = G(K₂|K₁) of G = G(K₂|K) with factor group G/U = G(K₁|K) = Syl₃C(K) = (3,3) must be the minimal subgroup of G with abelian factor group, i. e., must coincide with the commutator subgroup G' of G. Further, G is a 2-stage metabelian 3-group, since G' = G(K₂|K₁) = Syl₃C(K₁) is abelian, i.e., G'' = 1.

Warning:
Although taking G = G(K_n|K), for some integer n >= 3, similarly yields G' = G(K_n|K₁) and G/G' = (3,3), the Galois group G = G(K_n|K) of the n^th Hilbert 3-class field K_n of K over K is not a 2-stage metabelian 3-group any longer, in general, since G'' = G(K_n|K₂) != 1, if 3 divides the class number of K₂.

1.A. All known real quadratic examples:

In the following diagrams, we give the 2-stage metabelian 3-groups G = G(K₂|K) that occur for real quadratic fields K = Q(d^1/2) with discriminant 0 < d < 200000. We denote by G = G₁ >= G₂ >= ... >= G_i >= ... the descending central series of G with G_i+1 = [G_i,G]. In particular, G₂ = [G,G] is the commutator subgroup G' of G.

Types a.1 and a.2

(1) Group of maximal class
(e = 2)
G = G^(4,4)(alpha,beta,gamma)
in ZEF 2a(4,4)
with (alpha,beta,gamma) = (0,+-1,0),
resp. (alpha,beta,gamma) = (1,0,0),
of class 3 and of order 3⁴ = 81

				K₂
				\|
				F₃
				\|
				K₁
	/		/		\		\
N₁		N₂				N₃		N₄
	\		\		/		/
				K

				G₄ = 1
				\|
				G₃ = (3)
				\|
				G₂ = (3,3)
	/		/		\		\
H₁		H₂				H₃		H₄
	\		\		/		/
				G₁ = G

Minimal occurrence:
discriminant d = 32009, resp.
discriminant d = 72329
Principalization type of K in N₁,...,N₄:
a.1, (2,0,0,0), resp. a.2, (1,0,0,0) [5,6] ,
both designated by a in [3]

Remarks:
The principalization types (2,0,0,0) and (1,0,0,0) do not determine the associated Galois group G = G(K₂|K) uniquely.

Results discovered 2003/02/07:
However, according to top recent computations of the structure of Syl₃C(K₁) = G(K₂|K₁) = G₂ as (3,3) by Karim Belabas with the aid of PARI, G is determined uniquely for principalization type (1,0,0,0) and uniquely up to the sign of beta for principalization type (2,0,0,0).
Further, exactly the same diagram illustrates the descending central series G = G₁ >= G₂ >= ... for the real quadratic fields K with discriminants
d = 42817 of principalization type a.1, (2,0,0,0) [5,6] , resp.
d = 94636 of principalization type a.2, (1,0,0,0) [5,6] .

Results discovered 2003/02/27:
In fact, we now have further occurrences of this frequent case:
d = 103809, 114889, 130397, 142097, 151141, 153949, 172252, 173944, 184137, 189237,
according to [6] and computations of Karim Belabas with the aid of PARI.

For this frequent case, we obviously have generally
Syl₃(C(K₁)) = (3,3) <==>
G(K₂|K) in ZEF 2a(4,4) <==>
family principal factorization type (Delta1,Alpha1,Alpha1,Alpha1) <==>
principalization type either (1,0,0,0) or (2,0,0,0),
i. e., the full 3-class group of K capitulates in
three of the four unramified cyclic cubic extensions N₁,...,N₄ of K,
in the fourth one, say N₁, only a subgroup of type (3) capitulates
and this is either Norm_{N₁|K}(C(N₁)) for type a.2, (1,0,0,0)
or Norm_{N_j|K}(C(N_j)) with 2 <= j <= 4 for type a.1, (2,0,0,0).

Further, N₁,...,N₄ have uniformly 3-class number 3.

Type a.3

(2) Group of maximal class
(e = 2)
G = G^(6,6)(alpha,beta,gamma)
in ZEF 2b(6,6)
with (alpha,beta,gamma) =
(0,0,1) resp. (+-1,0,1),
of class 5 and of order 3⁶ = 729

				K₂
				\|
				F₅
				\|
				F₄
				\|
				F₃
				\|
				K₁
	/		/		\		\
N₁		N₂				N₃		N₄
	\		\		/		/
				K

				G₆ = 1
				\|
				G₅ = (3)
				\|
				G₄ = (3,3)
				\|
				G₃ = (3²,3)
				\|
				G₂ = (3²,3²)
	/		/		\		\
H₁		H₂				H₃		H₄
	\		\		/		/
				G₁ = G

Minimal and up to 10⁵ unique occurrence:
discriminant d = 62501
Principalization type of K in N₁,...,N₄:
a.3, (0,0,0,0) [5,6] ,
designated by a in [3]

Remarks:
Again, the principalization type (0,0,0,0) does not determine the associated Galois group G = G(K₂|K) uniquely.

Result discovered 2003/02/07:
However, according to a top recent computation of the structure of Syl₃C(K₁) = G(K₂|K₁) = G₂ as (9,9) by Karim Belabas with the aid of PARI, at least beta = 0, the class 5, and the order 729 of G are determined uniquely.

Result discovered 2003/02/27:
In fact, we now have two occurrences of this rare case:
d = 62501 and d = 152949,
according to [6] and a computation of Karim Belabas with the aid of PARI.
For these fields, I got family principal factorization type (Alpha1,Alpha1,Alpha1,Alpha1)
already in August 1991 at Winnipeg City [6] .

For this rare case, we obviously have generally
Syl₃(C(K₁)) = (9,9) <==>
G(K₂|K) in ZEF 2b(6,6) <==>
family principal factorization type (Alpha1,Alpha1,Alpha1,Alpha1) <==>
principalization type a.3, (0,0,0,0),
i. e., the full 3-class group of K capitulates in
all four unramified cyclic cubic extensions N₁,...,N₄ of K.

Further, exactly one of N₁,...,N₄ has 3-class number 9,
the others have 3-class number only 3.
This fact is due to formulas for the transfers ("Verlagerungen") and
shows that the last parameter of the group G must be gamma = 1.

1.B. All known imaginary quadratic examples:

In the following diagrams, we give the 2-stage metabelian 3-groups G = G(K₂|K) that occur for imaginary quadratic fields K = Q(d^1/2) with discriminant -50000 < d < 0. We denote by G = G₁ >= G₂ >= ... >= G_i >= ... the descending central series of G with G_i+1 = [G_i,G]. In particular, G₂ = [G,G] is the commutator subgroup G' of G.

Type D

(1) Group of second maximal class
(e = 3)
G = G^(4,5)((alpha,beta,gamma,delta),rho)
in ZEF 1a(4,5)
with (alpha,beta,gamma,delta) = (0,0,-1,1), rho = 0,
resp. (alpha,beta,gamma,delta) = (1,1,-1,1), rho = 0,
of class 3 and of order 3⁵ = 243

				K₂
				\|\|
				F₃
				\|
				K₁
	/		/		\		\
N₁		N₂				N₃		N₄
	\		\		/		/
				K

				G₄ = 1
				\|\|
				G₃ = (3,3)
				\|
				G₂ = (3,3,3)
	/		/		\		\
H₁		H₂				H₃		H₄
	\		\		/		/
				G₁ = G

Minimal occurrence:
discriminant d = -4027, resp.
discriminant d = -12131
Principalization type of K in N₁,...,N₄:
D.10, (1,1,2,3), resp. D.5, (1,2,1,2) [4,5] ,
both designated by D in [1]

Remarks:
(1,1,2,3) and (1,2,1,2) are the only two principalization types where the associated Galois group G = G(K₂|K) is uniquely determined.

Exactly the same diagram illustrates the descending central series G = G₁ >= G₂ >= ... for the imaginary quadratic fields K with discriminants
d = -8751, -19651, -21224, -22711, -24904, -26139, -28031, -28759 of principalization type D.10, (1,1,2,3) [4,5] , resp.
d = -19187, -20276, -20568, -24340, -26760 of principalization type D.5, (1,2,1,2) [4,5] .

The Hilbert 3-class field tower K = K₀ <= K₁ <= K₂ <= ... of all these fields terminates after 2 steps with K₂, i. e., 3 does not divide the class number of K₂. This was shown by Scholz and Taussky in [1] and, with a different proof, by Brink and Gold in [2] .

Results discovered 2003/04/19 - 22 and 2003/05/09:
According to [7] and computations of Karim Belabas with the aid of PARI, we now have further occurrences of this case:
d = -31639, -31999, -32968, -34088, -34507, -35367, -36807, -40299, -40692, -41015, -41583, -41671, -42423, -43192, -43307,
and:
d = -44004, -45835, -46587, -48052, -49128, -49812.

Type E

(2) Group of second maximal class
(e = 3)
G = G^(6,7)((alpha,beta,gamma,delta),rho)
in ZEF 2a(6,7)
with (alpha,beta,gamma,delta) = (0,0,+-1,1), rho = 0,
of class 5 and of order 3⁷ = 2187

				K₂
				\|
				F₅
				\|
				F₄
				\|\|
				F₃
				\|
				K₁
	/		/		\		\
N₁		N₂				N₃		N₄
	\		\		/		/
				K

				G₆ = 1
				\|
				G₅ = (3)
				\|
				G₄ = (3,3)
				\|\|
				G₃ = (3²,3,3)
				\|
				G₂ = (3²,3²,3)
	/		/		\		\
H₁		H₂				H₃		H₄
	\		\		/		/
				G₁ = G

Minimal occurrence:
discriminant d = -9748
Principalization type of K in N₁,...,N₄:
E.9, (1,1,3,2) or equivalently (1,2,1,3) [4,5] ,
designated by E in [1]

Remarks:
Here, the principalization type (1,1,3,2) does not determine the associated Galois group G = G(K₂|K) uniquely.
However, Scholz and Taussky [1] provide additional information for d = -9748 mentioning that a = 4 in the associated symbolic order X_a = (X^a,XY,Y²,3+3X+3X²) whence G' = G₂ = Z[X,Y]/X_a = (3²,3²,3), and this property determines G uniquely, up to the sign of gamma.

The Hilbert 3-class field tower K = K₀ <= K₁ <= K₂ <= ... of the field K = Q((-9748)^1/2) terminates after 2 steps with K₂, i. e., 3 does not divide the class number of K₂. That is the only claim by Scholz and Taussky in [1] and not a statement for all fields with associated symbolic order X_a. Heider and Schmithals erroneously remark in [5] that Scholz and Taussky proved G₄ = 1 for the fields K with associated symbolic order X_a, whereas they really have G₄ = (3,3). This led to a misinterpretation by Brink and Gold in [2] , where they construct a 3-stage metabelian 3-group that could possibly be the Galois group G(M|K) of an unramified cubic extension M of K₂ when K has the associated symbolic order X_a. But no explicit example is known up to now for a 3-class field tower of height greater than 2.

Results discovered 2003/01/14 - 18:
According to recent computations of the structure of Syl₃C(K₁) = G(K₂|K₁) = G₂ as (9,9,3) by Karim Belabas with the aid of PARI, exactly the same diagram illustrates the descending central series G = G₁ >= G₂ >= ... for the imaginary quadratic fields K with discriminants
d = -22395, -22443, -27640 of principalization type E.9, (1,1,3,2) ~ (1,2,1,3) [4] ,
with (alpha,beta,gamma,delta) = (0,0,+-1,1), rho = 0, resp.
d = -15544, -18555, -23683 of principalization type E.6, (1,1,2,2) [4,5] ,
with (alpha,beta,gamma,delta) = (1,-1,1,1), rho = 0, resp.
d = -16627 of principalization type E.14, (2,3,1,1) [4,5] ,
with (alpha,beta,gamma,delta) = (0,-1,+-1,1), rho = 0,
which are all designated by E in [1] .

Results discovered 2003/03/23 and 2003/05/24:
According to [7] and computations of Karim Belabas with the aid of PARI, we now have further occurrences of this case:
d = -31271, -34867, -37988, -39736, -42619, -42859, -43847, -45887, -48472, -48667.

More detailed, according to [7] , we finally have
d = -37988, -39736, -45887, -48472, -48667 of principalization type E.9, (1,1,3,2) ~ (1,2,1,3),
with (alpha,beta,gamma,delta) = (0,0,+-1,1), rho = 0,
d = -31271, -42859, -43847 of principalization type E.14, (2,3,1,1),
with (alpha,beta,gamma,delta) = (0,-1,+-1,1), rho = 0, and
d = -34867, -42619 of the newly discovered principalization type E.8, (1,2,3,1), the unique one with 3 fixed points (A,A,A,B) and
with (alpha,beta,gamma,delta) = (1,0,-1,1), rho = 0.

Type H

(3) Group of second maximal class
(e = 3)
G = G^(5,6)((alpha,beta,gamma,delta),rho)
in ZEF 1b(5,6) or ZEF 2a(5,6)
with (alpha,beta,gamma,delta) = (?,1,?,1), rho = 1
or (alpha,beta,gamma,delta) = (1,-1,-1,1), rho = 0,
of class 4 and of order 3⁶ = 729
with rho != beta - 1 or rho = 0

				K₂
				\|
				F₄
				\|\|
				F₃
				\|
				K₁
	/		/		\		\
N₁		N₂				N₃		N₄
	\		\		/		/
				K

				G₅ = 1
				\|
				G₄ = (3)
				\|\|
				G₃ = (3,3,3)
				\|
				G₂ = (3²,3,3)
	/		/		\		\
H₁		H₂				H₃		H₄
	\		\		/		/
				G₁ = G

Minimal occurrence:
discriminant d = -3896
Principalization type of K in N₁,...,N₄:
H.4, (2,1,1,1) [4,5] ,
designated by H in [1]

Remarks:
Again, the principalization type (2,1,1,1) does not determine the associated Galois group G = G(K₂|K) uniquely.

Results discovered 2003/01/14 and 2003/02/17:
However, according to recent computations of the structure of Syl₃C(K₁) = G(K₂|K₁) = G₂ as (9,3,3) by Karim Belabas with the aid of PARI and Claus Fieker with the aid of MAGMA, at least delta = 1, the class 4, and the order 729 of G are determined uniquely.

Exactly the same diagram illustrates the descending central series G = G₁ >= G₂ >= ... for the imaginary quadratic fields K with discriminants
d = -6583, -23428, -25447, -27355, -27991 of the same principalization type H.4, (2,1,1,1) [4,5] ,
designated by H in [1] .

Results discovered 2003/04/22 and 2003/05/09:
According to [7] and computations of Karim Belabas with the aid of PARI, we now have further occurrences of this case:
d = -36276, -37219, -37540, -39819, -41063,
and:
d = -43827, -46551.

Type G.19

(4) Group of second maximal class
(e = 3)
G = G^(5,6)((alpha,beta,gamma,delta),rho)
in ZEF 1b(5,6)
with (alpha,beta,gamma,delta) = (?,0,?,1), rho = -1,
of class 4 and of order 3⁶ = 729
with rho = beta - 1 != 0

				K₂
				\|
				F₄
				\|\|
				F₃
				\|
				K₁
	/		/		\		\
N₁		N₂				N₃		N₄
	\		\		/		/
				K

				G₅ = 1
				\|
				G₄ = (3)
				\|\|
				G₃ = (3,3,3)
				\|
				G₂ = (3,3,3,3)
	/		/		\		\
H₁		H₂				H₃		H₄
	\		\		/		/
				G₁ = G

Minimal occurrence:
discriminant d = -12067
Principalization type of K in N₁,...,N₄:
G.19, (2,1,4,3) [4,5] ,
designated by G in [1]

Remarks:
As above, the principalization type (2,1,4,3) does not determine the associated Galois group G = G(K₂|K) uniquely.

Result discovered 2003/01/14:
But according to the recent computation of the structure of Syl₃C(K₁) = G(K₂|K₁) = G₂ as (3,3,3,3) by Karim Belabas with the aid of PARI, at least beta = 0, delta = 1, rho = -1, the class 4, and the order 729 of G are determined uniquely.

Result discovered 2003/05/10:
According to [7] , we now have another occurrence of this rare case:
d = -49924.

Type F

(5) Group of lower than second maximal class
(e = 5)
G = G^(6,9)((alpha,beta,gamma,delta),rho)
in ZEF 1a(6,9)
with (alpha,beta,gamma,delta) = (?,?,0,0), rho = 0,
of class 5 and of order 3⁹ = 19683

				K₂
				\|\|
				F₅
				\|\|
				F₄
				\|\|
				F₃
				\|
				K₁
	/		/		\		\
N₁		N₂				N₃		N₄
	\		\		/		/
				K

				G₆ = 1
				\|\|
				G₅ = (3,3)
				\|\|
				G₄ = (3,3,3,3)
				\|\|
				G₃ = (3²,3²,3,3)
				\|
				G₂ = (3²,3²,3²,3)
	/		/		\		\
H₁		H₂				H₃		H₄
	\		\		/		/
				G₁ = G

Minimal occurrence:
discriminant d = -27156
Principalization type of K in N₁,...,N₄:
F.11, (1,3,2,1) [4] ,
designated by F in [1]

Remarks:
Again, the principalization type (1,3,2,1) does not determine the associated Galois group G = G(K₂|K) uniquely.

Result discovered 2003/02/13:
However, according to a top recent computation of the structure of Syl₃C(K₁) = G(K₂|K₁) = G₂ as (9,9,9,3) by Claus Fieker with the aid of MAGMA, at least gamma = 0, delta = 0, rho = 0, the class 5, and the order 19683 of G are determined uniquely.

Results discovered 2003/03/23, 2003/05/27 - 31, and 2003/06/10:
According to [7] and computations of Karim Belabas with the aid of PARI, we now have other occurrences of this rare case:
d = -31908, -135587 of the newly discovered principalization type F.12, (2,1,3,1) ~ (3,2,1,1).

According to the supplements section of [7] , we finally have other occurrences of this rare case:
d = -67480, -104627 of the newly discovered principalization type F.13, (2,1,1,3),
and d = -124363 of the newly discovered principalization type F.7, (2,1,1,2).

Results discovered 2003/09/16:
According to the supplements section of [7] and computations of Karim Belabas with the aid of PARI, we now have further occurrences of this case:
d = -160403, -167064, -184132 and d = -189959.

Type G.16 and a Variant of Type H

(6) Group of second maximal class
(e = 3)
G = G^(7,8)((alpha,beta,gamma,delta),rho)
in ZEF 2a(7,8) or ZEF 2b(7,8)
with (alpha,beta,gamma,delta) = (1,0,0,1), rho = 0,
or (alpha,beta,gamma,delta) = (?,0,?,1), rho = +-1,
of class 6 and of order 3⁸ = 6561

				K₂
				\|
				F₆
				\|
				F₅
				\|
				F₄
				\|\|
				F₃
				\|
				K₁
	/		/		\		\
N₁		N₂				N₃		N₄
	\		\		/		/
				K

				G₇ = 1
				\|
				G₆ = (3)
				\|
				G₅ = (3,3)
				\|
				G₄ = (3²,3)
				\|\|
				G₃ = (3²,3²,3)
				\|
				G₂ = (3³,3²,3)
	/		/		\		\
H₁		H₂				H₃		H₄
	\		\		/		/
				G₁ = G

Minimal occurrence:
discriminant d = -17131
Principalization type of K in N₁,...,N₄:
G.16, (1,2,4,3) [4,5] ,
designated by G in [1]

Remarks:
As above, the principalization type (1,2,4,3) does not determine the associated Galois group G = G(K₂|K) uniquely.

Results discovered 2003/02/17:
However, according to a top recent computation of the structure of Syl₃C(K₁) = G(K₂|K₁) = G₂ as (27,9,3) by Claus Fieker with the aid of MAGMA, at least beta = 0, delta = 1, the class 6, and the order 6561 of G are determined uniquely.

Exactly the same diagram illustrates the descending central series G = G₁ >= G₂ >= ... for the imaginary quadratic fields K with discriminants
d = -24884, -28279 of the same principalization type G.16, (1,2,4,3) ~ (2,1,3,4) [4] ,
with (alpha,beta,gamma,delta) = (1,0,0,1), rho = 0 or (alpha,beta,gamma,delta) = (?,0,?,1), rho = +-1,
designated by G in [1] , resp.
d = -21668 of principalization type H.4, (2,1,1,1) [4] ,
with (alpha,beta,gamma,delta) = (1,-1,-1,1), rho = 0 or (alpha,beta,gamma,delta) = (?,-1,?,1), rho = +-1,
designated by H in [1] .

Results discovered 2003/03/23:
According to [7] and computations of Karim Belabas with the aid of PARI, we now have further occurrences of this case:
d = -34027 of principalization type H.4, (2,1,1,1) and d = -35539 of principalization type G.16, (1,2,4,3) ~ (2,1,3,4).

A Variant of Type G.19

(7) Group of lower than second maximal class
(e = 5)
G = G^(7,10)((alpha,beta,gamma,delta),rho)
in ZEF 2a(7,10) or ZEF 1b(7,10)
with (alpha,beta,gamma,delta) = (0,?,?,0), rho = 0,
or (alpha,beta,gamma,delta) = (?,?,?,0), rho = +1,
of class 6 and of order 3¹⁰ = 59049

				K₂
				\|
				F₆
				\|\|
				F₅
				\|\|
				F₄
				\|\|
				F₃
				\|
				K₁
	/		/		\		\
N₁		N₂				N₃		N₄
	\		\		/		/
				K

				G₇ = 1
				\|
				G₆ = (3)
				\|\|
				G₅ = (3,3,3)
				\|\|
				G₄ = (3²,3,3,3)
				\|\|
				G₃ = (3²,3²,3²,3)
				\|
				G₂ = (3³,3²,3²,3)
	/		/		\		\
H₁		H₂				H₃		H₄
	\		\		/		/
				G₁ = G

Minimal and up to |d| < 150000 unique occurrence:
discriminant d = -96827
Principalization type of K in N₁,...,N₄:
G.19, (2,1,4,3) [7] ,
designated by G in [1]

Remarks:
As above, the principalization type (2,1,4,3) does not determine the associated Galois group G = G(K₂|K) uniquely.

Result discovered 2003/05/29:
However, according to a top recent computation of the structure of Syl₃C(K₁) = G(K₂|K₁) = G₂ as (27,9,9,3) by Karim Belabas with the aid of PARI, at least delta = 0, the class 6, and the order 59049 of G are determined uniquely.

Result discovered 2003/09/16:
According to the supplements section of [7] and a computation of Karim Belabas with the aid of PARI, we now have a further occurrence of this case:
d = -156452.

A Variant of Type G.16

(8) Group of lower than second maximal class
(e = 5)
G = G^(7,10)((alpha,beta,gamma,delta),rho)
in ZEF 1b(7,10)
or (alpha,beta,gamma,delta) = (?,?,?,0), rho = -1,
of class 6 and of order 3¹⁰ = 59049

				K₂
				\|
				F₆
				\|\|
				F₅
				\|\|
				F₄
				\|\|
				F₃
				\|
				K₁
	/		/		\		\
N₁		N₂				N₃		N₄
	\		\		/		/
				K

				G₇ = 1
				\|
				G₆ = (3)
				\|\|
				G₅ = (3,3,3)
				\|\|
				G₄ = (3²,3,3,3)
				\|\|
				G₃ = (3²,3²,3²,3)
				\|
				G₂ = (3²,3²,3²,3²)
	/		/		\		\
H₁		H₂				H₃		H₄
	\		\		/		/
				G₁ = G

Minimal and up to |d| < 150000 unique occurrence:
discriminant d = -135059
Principalization type of K in N₁,...,N₄:
G.16, (1,2,4,3) [7] ,
designated by G in [1]

Remarks:
As above, the principalization type (1,2,4,3) does not determine the associated Galois group G = G(K₂|K) uniquely.

Result discovered 2003/06/10:
However, according to a top recent computation of the structure of Syl₃C(K₁) = G(K₂|K₁) = G₂ as (9,9,9,9) by Karim Belabas with the aid of PARI, at least delta = 0, rho = -1, the class 6, and the order 59049 of G are determined uniquely.

Results discovered 2003/09/16:
According to the supplements section of [7] and computations of Karim Belabas with the aid of PARI, we now have further occurrences of this case:
d = -185747 and d = -186483.

References:

[1] Arnold Scholz und Olga Taussky,
Die Hauptideale der kubischen Klassenkörper
imaginär quadratischer Zahlkörper,
J. reine angew. Math.171 (1934), 19 - 41

[2] James R. Brink and Robert Gold,
Class field towers of imaginary quadratic fields,
manuscripta math. 57 (1987), 425 - 450

[3] Brigitte Nebelung,
Klassifikation metabelscher 3-Gruppen
mit Faktorkommutatorgruppe vom Typ (3,3)
und Anwendung auf das Kapitulationsproblem,
Inauguraldissertation, K�ln, 1989

[4] Daniel C. Mayer,
Principalization in complex S₃-fields,
Congressus Numerantium 80 (1991), 73 - 87

[5] Franz-Peter Heider und Bodo Schmithals,
Zur Kapitulation der Idealklassen in unverzweigten primzyklischen Erweiterungen,
J. reine angew. Math. 336 (1982), 1 - 25

[6] Daniel C. Mayer,
List of discriminants d_L<200000 of totally real cubic fields L,
arranged according to their multiplicities m and conductors f,
1991, Dept. of Comp. Sci., Univ. of Manitoba

[7] Daniel C. Mayer,
Principalization in Unramified Cubic Extensions
of Quadratic Fields with Discriminant -50000 < d < 0,
Univ. Graz, Computer Centre, 2003

2. A modern point of view.

Aïssa Derhem [2], in Casablanca, Maroc, was the first to observe that Nebelung's results [1], can be applied equally well to the following questions that arise for the absolute 3-genus field K* = (K|Q)* of a cyclic cubic number field K, whose conductor f has exactly 2 prime divisors, whence K* is a bicyclic bicubic field:

1. to determine the absolute Galois group G(K*₁|Q) of the 1^st Hilbert 3-class field K*₁ of K* over Q,
2. to find the structure of the 3-class group Syl₃C(K*) of K*.

				K*₁
				\|
				K*
				\|
				Q

G(K₁\|K₁) = 1
\|
G' = G(K₁\|K) = Syl₃C(K*)	----
\|		G/G' = G(K*\|Q) = (3,3)
G = G(K*₁\|Q)	----

This application is due to genus field theory, since K* is the maximal abelian 3-extension of Q that is unramified over K and thus the subgroup U = G(K*₁|K*) of G = G(K*₁|Q) with factor group G/U = G(K*|Q) = (3,3) must be the minimal subgroup of G with abelian factor group, i. e., must coincide with the commutator subgroup G' of G. Further, G is a 2-stage metabelian 3-group, since G' = G(K*₁|K*) = Syl₃C(K*) is abelian, i.e., G'' = 1.

2.A. All known cyclic cubic examples:

In the following diagrams, we give all 2-stage metabelian 3-groups G = G(K*₁|Q) that occured for cyclic cubic fields K with 2-prime conductors f < 10⁵ [3]. We denote by G = G₁ >= G₂ >= ... >= G_i >= ... the descending central series of G with G_i+1 = [G_i,G]. In particular, G₂ = [G,G] is the commutator subgroup G' of G.
The absolute 3-genus field K* is the compositum K* = k*k~ of k and k~, the cyclic cubic fields with the two primeconductors dividing f. Hence K* is a bicyclic bicubic field that contains yet another cyclic cubic field K~ with conductor f.

Group of maximal class
(e = 2)
G = (3,3)
in ZEF(2,2)
of class 1 and of order 3² = 9

				K₁ = K
	/		/		\		\
k		K				K~		k~
	\		\		/		/
				Q

				G₂ = 1
	/		/		\		\
h		H				H~		h~
	\		\		/		/
				G

Minimal occurrence:
f = 63 = 3²*7
3-class numbers of K and K~:
(3,3)

Group of maximal class
(e = 2)
G = G⁽³⁾(alpha,beta,gamma)
in ZEF 1a(3,3)
with (alpha,beta,gamma) = (0,0,0),
of class 2 and of order 3³ = 27

				K*₁
				\|
				K*
	/		/		\		\
k		K				K~		k~
	\		\		/		/
				Q

				G₃ = 1
				\|
				G₂ = (3)
	/		/		\		\
h		H				H~		h~
	\		\		/		/
				G

Minimal occurrence:
f = 657 = 3²*73
3-class numbers of K and K~:
(9,9)

Group of maximal class
(e = 2)
G = G⁽⁴⁾(alpha,beta,gamma)
in ZEF 2a(4,4)
with (alpha,beta,gamma) = (0,1,0),
of class 3 and of order 3⁴ = 81

				K*₁
				\|
				F₃
				\|
				K*
	/		/		\		\
k		K				K~		k~
	\		\		/		/
				Q

				G₄ = 1
				\|
				G₃ = (3)
				\|
				G₂ = (3,3)
	/		/		\		\
h		H				H~		h~
	\		\		/		/
				G

Minimal occurrence:
f = 2439 = 3²*271
3-class numbers of K and K~:
(9,9)

Group of second maximal class
(e = 3)
G = G^(4,5)((alpha,beta,gamma,delta),rho)
in ZEF 1a(4,5)
with (alpha,beta,gamma,delta) = (0,0,0,0), rho = 0,
of class 3 and of order 3⁵ = 243

				K*₁
				\|\|
				F₃
				\|
				K*
	/		/		\		\
k		K				K~		k~
	\		\		/		/
				Q

				G₄ = 1
				\|\|
				G₃ = (3,3)
				\|
				G₂ = (3,3,3)
	/		/		\		\
h		H				H~		h~
	\		\		/		/
				G

Minimal occurrence:
f = 4711 ("Eau de Cologne") = 7*673
3-class numbers of K and K~:
(27,27)

Group of second maximal class
(e = 3)
G = G^(5,6)((alpha,beta,gamma,delta),rho)
in ZEF 1b(5,6) resp. ZEF 2a(5,6)
with (alpha,beta,gamma,delta) = (0,0,0,0), rho = 1
resp. rho = 0,
of class 4 and of order 3⁶ = 729
with rho != beta - 1 resp. rho = 0

				K*₁
				\|
				F₄
				\|\|
				F₃
				\|
				K*
	/		/		\		\
k		K				K~		k~
	\		\		/		/
				Q

				G₅ = 1
				\|
				G₄ = (3)
				\|\|
				G₃ = (3,3,3)
				\|
				G₂ = (3²,3,3)
	/		/		\		\
h		H				H~		h~
	\		\		/		/
				G

Minimal occurrence:
f = 5383 = 7*769 resp.
f = 41977 = 13*3229
3-class numbers of K and K~:
(27,27) resp. (81,27)

Group of second maximal class
(e = 3)
G = G^(5,6)((alpha,beta,gamma,delta),rho)
in ZEF 1b(5,6)
with (alpha,beta,gamma,delta) = (0,0,0,0), rho = -1,
of class 4 and of order 3⁶ = 729
with rho = beta - 1 != 0

				K*₁
				\|
				F₄
				\|\|
				F₃
				\|
				K*
	/		/		\		\
k		K				K~		k~
	\		\		/		/
				Q

				G₅ = 1
				\|
				G₄ = (3)
				\|\|
				G₃ = (3,3,3)
				\|
				G₂ = (3,3,3,3)
	/		/		\		\
h		H				H~		h~
	\		\		/		/
				G

Minimal occurrence:
f = 21763 = 7*3109
3-class numbers of K and K~:
(27,27)

Remark:
Here, the difference to f = 5383 is that we have G₂ = (3,3,3,3) and rho = -1 instead of G₂ = (3²,3,3) and rho = +1, which implies different structures of Syl₃C(K*) = G(K*₁|K*) = G₂.

Group of second maximal class
(e = 3)
G = G^(6,7)((alpha,beta,gamma,delta),rho)
in ZEF 2b(6,7)
with (alpha,beta,gamma,delta) = (0,0,0,0), rho = +-1,
of class 5 and of order 3⁷ = 2187

				K*₁
				\|
				F₅
				\|
				F₄
				\|\|
				F₃
				\|
				K*
	/		/		\		\
k		K				K~		k~
	\		\		/		/
				Q

				G₆ = 1
				\|
				G₅ = (3)
				\|
				G₄ = (3,3)
				\|\|
				G₃ = (3²,3,3)
				\|
				G₂ = (3²,3²,3)
	/		/		\		\
h		H				H~		h~
	\		\		/		/
				G

Minimal occurrence:
f = 68857 = 37*1861
3-class numbers of K and K~:
(81,27)

Group of second maximal class
(e = 3)
G = G^(7,8)((alpha,beta,gamma,delta),rho)
in ZEF 2b(7,8)
with (alpha,beta,gamma,delta) = (0,0,0,0), rho = +-1,
of class 6 and of order 3⁸ = 6561

				K*₁
				\|
				F₆
				\|
				F₅
				\|
				F₄
				\|\|
				F₃
				\|
				K*
	/		/		\		\
k		K				K~		k~
	\		\		/		/
				Q

				G₇ = 1
				\|
				G₆ = (3)
				\|
				G₅ = (3,3)
				\|
				G₄ = (3²,3)
				\|\|
				G₃ = (3²,3²,3)
				\|
				G₂ = (3³,3²,3)
	/		/		\		\
h		H				H~		h~
	\		\		/		/
				G

Minimal occurrence:
f = 36667 = 37*991
3-class numbers of K and K~:
(243,27)

Group of lower than second maximal class
(e = 4)
G = G^(6,8)((alpha,beta,gamma,delta),rho)
in ZEF 2a(6,8)
with (alpha,beta,gamma,delta) = (0,0,0,0), rho = 0,
of class 5 and of order 3⁸ = 6561

				K*₁
				\|
				F₅
				\|\|
				F₄
				\|\|
				F₃
				\|
				K*
	/		/		\		\
k		K				K~		k~
	\		\		/		/
				Q

				G₆ = 1
				\|
				G₅ = (3)
				\|\|
				G₄ = (3,3,3)
				\|\|
				G₃ = (3²,3,3,3)
				\|
				G₂ = (3²,3²,3,3)
	/		/		\		\
h		H				H~		h~
	\		\		/		/
				G

Minimal occurrence:
f = 42127 = 103*409
3-class numbers of K and K~:
(243,81)

References:

[1] Brigitte Nebelung,
Klassifikation metabelscher 3-Gruppen
mit Faktorkommutatorgruppe vom Typ (3,3)
und Anwendung auf das Kapitulationsproblem,
Inauguraldissertation, K�ln, 1989

[2] Aïssa Derhem,
Sur les corps cubiques cycliques
de conducteur divisible par deux premiers,
Casablanca, 2002

[3] Daniel C. Mayer,
Class Numbers and Principal Factorizations of Families
of Cyclic Cubic Fields with Discriminant d < 10¹⁰,
Univ. Graz, Computer Centre, 2002

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