1. Definition of Principalization Types.Let K be a base field with 3class rank 2 andwith 3class group Syl_{3}(C(K)) of 3elementary abelian type (3,3). Denote by K_{1} the 1^{st} Hilbert 3class field of K. According to the Artin Reciprocity Law of Class Field Theory, the 3class group Syl_{3}(C(K)) = (3,3) of K is isomorphic to the relative automorphism group Gal(K_{1}K) of K_{1} over K. In particular, the 4 cyclic subgroups C_{i} (1 <= i <= 4) of Syl_{3}(C(K)) are mapped to the Galois groups Gal(K_{1}N_{i}) = M_{i} of 4 unramified cyclic cubic extensions N_{i} of K. In fact, the C_{i} = Norm(N_{i}K)Syl_{3}(C(N_{i})) are relative norms of 3class groups. The following diagrams illustrate the Galois correspondence and the Artin isomorphism:
Now we consider the class extension homomorphisms j(N_{i}K): Syl_{3}(C(K)) > Syl_{3}(C(N_{i})). We say an ideal class of K that is mapped to the principal class 1 of N_{i} by j(N_{i}K) principalizes or becomes principal or capitulates in N_{i}. According to Hilbert's Theorem 94, none of the principalization kernels Ker j(N_{i}K) is trivial, since there is an isomorphism to the unit cohomology of N = N_{i}: Ker j(NK) = P_{N}^{ < S > }/P_{K} = E_{NK}/U_{N}^{S1} != 1, where Gal(NK) = < S > and E_{NK} denotes the intersection of U_{N} with Ker Norm(NK). According to the Hilbert/Artin/Furtwängler Principal Ideal Theorem, we have complete principalization in the Hilbert 3class field K_{1} = N_{0}: Ker j(N_{0}K) = Syl_{3}(C(K)) = C_{0}. Thus, there are 5 possibilities for each Ker j(N_{i}K), C_{0},...,C_{4}, and if K is a quadratic field we define a Natural Principalization Type (k(1),...,k(4)), ordering the N_{i} (1 <= i <= 4),which have dihedral absolute groups Gal(N_{i}Q) = D(6), by increasing regulators of their absolute cubic subfields L_{i}: for each index 0 <= i <= 4 there exists a unique index 0 <= k(i) <= 4 such that Ker j(N_{i}K) = C_{k(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(d^{1/2}) with discriminant 50000 < d < 0 and 3class group of type (3,3). These recent computations of 2003 for 42 new cases in the range 50000 < d < 30000 [7] extend our own results for 22 fields with 30000 < d < 20000 of 1989 [5] and the 13 examples with 20000 < d < 0 of Heider and Schmithals in 1982 [2] . As a supplement, we give some singular cases with associated 2stage metabelian 3groups of exceptionally high order in the range 200000 < d < 50000 [8] . 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 2stage metabelian 3groups, we published several top recent results (discovered between January and August, 2003) for quadratic fields K whose Sylow 3subgroup Syl_{3}(C(K)) of the class group C(K), i. e., the 3class group, is of type (3,3). We denoted by G = G_{1} > G_{2} > ... > G_{i} > ... > G_{m} = 1 the descending central series of class m  1 of G with G_{i+1} = [G_{i},G]. In particular, G_{2} = [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 m1 and the order 3^{n} of the 2stage metabelian 3group G = G(K_{2}K) in ZEF(m,n) of the 2nd Hilbert 3class field K_{2} of an arbitrary base field K with 3class group Syl_{3}(C(K)) = (3,3) over K from the structure of the (abelian) commutator subgroup G' = G_{2} = Gal(K_{2}K_{1}) = Syl_{3}(C(K_{1})) which coincides with the 3class group of the 1st Hilbert 3class field K_{1} 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 3rank of G' is r <= 2 THEOREM 2. G is of second maximal class ( n = m + 1, resp. e = 3 ) <==> the 3rank 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 3rank 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 [4] . Here, we only give a coarse connection: Denote by 0 <= f <= 4 the number of those unramified cubic extensions of K where the full 3class 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 [1] 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 2stage metabelian 3groups G = G(K_{2}  K)for the 2^{nd} Hilbert 3class field K_{2} of an algebraic number field K with 3class group Syl_{3}(C(K)) of type (3,3) and with a certain principalization type (k(1),...,k(4)) in the four unramified cyclic cubic extensions N_{1},...,N_{4} of K. However, our concrete numerical results [7] 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 m1 and order 3^{n} of G = G(K_{2}  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 [5] . We always give a canonical representative (CR) of the type's equivalence class (S_{4}orbit), the year of the concrete numerical realization of the type with a reference, the first discriminant d_{K} 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 S_{4}, an ideal of polynomials in Z[X,Y], called the associated symbolic order (SO) in [1] , the structure of the commutator subgroup G' = G_{2}, the exponents in defining relations for the group's generators (RE), and the set ZEF(m,n), defined in [4] , to which the group G belongs. Finally, we note that the family of 3class numbers (h_{1},...,h_{4}) of the absolute cubic subfields L_{1},...,L_{4} of the normal S_{3}fields N_{1},...,N_{4} between K_{1} and K also permits partial conclusions concerning the group G.





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