1. Definition of Principalization Types.Let K be a base field with 3-class rank 2 and
with 3-class group Syl3(C(K)) of 3-elementary abelian type (3,3).
Denote by K1 the 1st Hilbert 3-class field of K.
According to the Artin Reciprocity Law of Class Field Theory,
the 3-class group Syl3(C(K)) = (3,3) of K is isomorphic to the
relative automorphism group Gal(K1|K) of K1 over K.
In particular, the 4 cyclic subgroups Ci (1 <= i <= 4) of Syl3(C(K)) are mapped to the
Galois groups Gal(K1|Ni) = Mi of 4 unramified cyclic cubic extensions Ni of K.
In fact, the Ci = Norm(Ni|K)Syl3(C(Ni)) are relative norms of 3-class groups.
The following diagrams illustrate the Galois correspondence and the Artin isomorphism:
Now we consider the class extension homomorphisms j(Ni|K): Syl3(C(K)) --> Syl3(C(Ni)).
We say an ideal class of K that is mapped to the principal class 1 of Ni by j(Ni|K)
principalizes or becomes principal or capitulates in Ni.
According to Hilbert's Theorem 94, none of the principalization kernels Ker j(Ni|K) is trivial,
since there is an isomorphism to the unit cohomology of N = Ni:
Ker j(N|K) = PN < S > /PK = EN|K/UNS-1 != 1,
where Gal(N|K) = < S > and EN|K denotes the intersection of UN with Ker Norm(N|K).
According to the Hilbert/Artin/Furtwängler Principal Ideal Theorem,
we have complete principalization in the Hilbert 3-class field K1 = N0:
Ker j(N0|K) = Syl3(C(K)) = C0.
Thus, there are 5 possibilities for each Ker j(Ni|K), C0,...,C4,
and if K is a quadratic field we define a Natural Principalization Type (k(1),...,k(4)),
ordering the Ni (1 <= i <= 4),which have dihedral absolute groups Gal(Ni|Q) = D(6),
by increasing regulators of their absolute cubic subfields Li:
for each index 0 <= i <= 4 there exists a unique index 0 <= k(i) <= 4
such that Ker j(Ni|K) = Ck(i) (in particular, always k(0) = 0).
An Important Application as Example.
In the following tables, we give the natural principalization type (k(1)...k(4))
of all imaginary quadratic fields K = Q(d1/2) with discriminant -50000 < d < 0 and 3-class group of type (3,3).
These recent computations of 2003 for 42 new cases in the range -50000 < d < -30000 
extend our own results for 22 fields with -30000 < d < -20000 of 1989 
and the 13 examples with -20000 < d < 0 of Heider and Schmithals in 1982  .
As a supplement, we give some singular cases
with associated 2-stage metabelian 3-groups of exceptionally high order
in the range -200000 < d < -50000  .
Exceptional cases are printed in boldface.
2. The Connection between Class, Order, and Commutator Subgroup.In the preceding section and in our previous communication,
concerning the descending central series of 2-stage metabelian 3-groups,
we published several top recent results (discovered between January and August, 2003) for quadratic fields K
whose Sylow 3-subgroup Syl3(C(K)) of the class group C(K), i. e., the 3-class group, is of type (3,3).
We denoted by G = G1 > G2 > ... > Gi > ... > Gm = 1 the descending central series of class m - 1 of G with Gi+1 = [Gi,G].
In particular, G2 = [G,G] is the commutator subgroup G' of G.
Now it is adequate to give the solution of a very fundamental problem:
How do we get the class m-1 and the order 3n
of the 2-stage metabelian 3-group G = G(K2|K) in ZEF(m,n) of the 2nd Hilbert 3-class field K2
of an arbitrary base field K with 3-class group Syl3(C(K)) = (3,3) over K
from the structure of the (abelian) commutator subgroup
G' = G2 = Gal(K2|K1) = Syl3(C(K1))
which coincides with the 3-class group of the 1st Hilbert 3-class field K1 of K:
1. There are bounds for the order in dependence on the class:
m <= n <= 2m - 3
2. Since there are only upper triangular entries in an (m,n)-diagram, we use an ordering parameter
e = n + 2 - m ( >= 2 )
to measure the distance from the diagonal ( e = 2 )
3. Then we have an almost unique correspondence between the structure of G' and the pair (m,n), expressed in the following (easily extensible) table .
Only the red pairs (m,n) are exceptional, since they correspond to 2 possible structures of G'.
In particular, we get 3 theorems concerning important cases:
THEOREM 1. G is of maximal class ( n = m, resp. e = 2 ) <==> the 3-rank of G' is r <= 2
THEOREM 2. G is of second maximal class ( n = m + 1, resp. e = 3 ) <==> the 3-rank of G' is r = 3 or G' = (3,3,3,3)
THEOREM 3. G is of lower than second maximal class ( n >= m + 2, resp. e >= 4 ) <==> the 3-rank of G' is r = 4 and the order of G' is >= 243
3. The Connection between Principalization Types, Class, and Order.The next question is: How are the principalization types related to the parameter e and the pairs (m,n) ?
The fine (necessary and sufficient) conditions have been determined by Brigitte Nebelung  .
Here, we only give a coarse connection:
Denote by 0 <= f <= 4 the number of those unramified cubic extensions of K
where the full 3-class group Syl_3(C(K)) = (3,3) of K becomes principal (i. e., capitulates).
Then we have 2 theorems:
THEOREM (i) G is of maximal class ( n = m, resp. e = 2 ) <==>
f = 4, i. e., principalization type (0,0,0,0) or
f = 3, i. e., principalization type (1,0,0,0) or (2,0,0,0) or
f = 0 and the principalization type is (1,1,1,1) (which implies n = 3 )
THEOREM (ii) G is of second maximal class or lower ( n >= m + 1, resp. e >= 3 ) <==>
f <= 2 but the principalization type is different from (1,1,1,1)
4. Peculiarities of Quadratic Fields.The last question is, what happens, if K is a quadratic field ?
Well, according to Scholz and Taussky  we have 2 theorems:
THEOREM 1. Principalization type (1,1,1,1) (denoted as Type "A" by S. & T.) is impossible for quadratic fields K
THEOREM 2. f = 0 for any imaginary quadratic field K
Example: d = -28031 ==> f = 0 and p.t. != (1,1,1,1) ==> e != 2 ==> r >= 3 ==> G' != (9,3). In fact, G' = (3,3,3).
5. Summary of recently (2003) discovered connections.In general, Nebelung's results still admit a wide range of 2-stage metabelian 3-groups G = G(K2 | K)
for the 2nd Hilbert 3-class field K2 of an algebraic number field K
with 3-class group Syl3(C(K)) of type (3,3)
and with a certain principalization type (k(1),...,k(4)) in the four unramified cyclic cubic extensions N1,...,N4 of K.
However, our concrete numerical results  discovered in 2003 for quadratic base fields K show that, in this special case,
a principalization type (k(1),...,k(4)) uniquely determines the class m-1 and order 3n of G = G(K2 | K),
except in the sections "F","G", and "H", where two possibilities arise.
(Section "A" is impossible for quadratic base fields K and
sections "B" and "C" cannot occur at all, for group theoretic reasons.)
In the following table, the types (k(1),...,k(4))) are arranged into sections, according to [1,4] ,
and they are numbered similarly as in  .
We always give a canonical representative (CR) of the type's equivalence class (S4-orbit),
the year of the concrete numerical realization of the type with a reference,
the first discriminant dK of a quadratic field K with that type,
the number of fixed points (FP),
the occupation numbers (ON) (telling how often each of the digits 1,2,3,4 appears in the representative),
the cardinality (#) of the type's orbit under the operation of S4,
an ideal of polynomials in Z[X,Y], called the associated symbolic order (SO) in  ,
the structure of the commutator subgroup G' = G2,
the exponents in defining relations for the group's generators (RE),
and the set ZEF(m,n), defined in  , to which the group G belongs.
Finally, we note that the family of 3-class numbers (h1,...,h4) of the absolute cubic subfields L1,...,L4
of the normal S3-fields N1,...,N4 between K1 and K
also permits partial conclusions concerning the group G.
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