The present contribution to the centennial festival with title "Memorial 2009" in honour of my invaluable academic advisor, the number theorist Alexander Aigner, is devoted to the fascinating realm of unramified cyclic cubic extensions NK of real quadratic number fields K with elementary abelian bicyclic 3class group Cl_{3}(K) of type (3,3) (i. e., isomorphic with the direct product of two cyclic groups C_{3} of order 3), and the numerous interesting arithmetical phenomena arising in this context. 
According to the combination of Artin's reciprocity law with the Galois correspondence, these real quadratic number fields K have exactly four nonisomorphic unramified cyclic cubic extensions N_{1}, N_{2}, N_{3}, N_{4} sharing the same discriminant d^{3}, where d denotes the discriminant of the maximal order of K. This simple relation between the discriminants is due to the relative conductor f = 1 of unramified extensions. Each of these unramified cyclic cubic relative extensions N_{i}K with relative group Gal(N_{i}K)=C_{3} is an absolute Galois extension with dihedral group Gal(N_{i}Q)=D(6) of order 6 over the rational number field Q. Therefore, by Galois theory, N_{i} contains three conjugate (and thus isomorphic) nonGalois absolutely cubic fields, which we shall identify and designate by L_{i}, since all their arithmetical invariants coincide. By a quadruplet of totally real cubic number fields we understand the family of these four nonisomorphic representatives of isomorphism classes L_{1}, L_{2}, L_{3}, L_{4}, all of whose members share the same fundamental discriminant d (equal to the quadratic discriminant), and which is in onetoone correspondence with the quadruplet N_{1}, N_{2}, N_{3}, N_{4}. 
We consider three closely related arithmetical problems for these quadruplets, the determination of
Here we mention that a complete, 2dimensional capitulation is equivalent with a principal factorization type Alpha_{1}, whereas a partial, 1dimensional capitulation is equivalent with a principal factorization type Delta_{1}. Type Alpha_{1} can be identified by the connection with the cohomology of the units, but for the identification of type Delta_{1} the capitulation kernel must be determined. The second and third problem can be solved by adequate theoretical statements, which we have developed in [7]. 
The earlier investigators Heider and Schmithals [2] (1982) used the table of Angell [1] to find the discriminants and units. For our first extension [4] in August 1991, which was mainly devoted to the principal factorization types of ramified fields, we computed the units with the aid of Voronoi's algorithm. 15 years later we started our extensive Real Capitulation Project in January 2006. In our second extension [5] (February 2006), we utilized the table of Ennola and Turunen [3] to find the discriminants and generating polynomials, performed Tschirnhausen transformations to obtain tracefree polynomials, and executed Voronoi's algorithm to get units. However, we desired an independent verification of and a break through beyond the range of Ennola and Turunen. In April 2006, we therefore computed all totally real cubic fields L with fundamental discriminants 0 < d < 10^{6}, i. e., only those whose normal fields N are unramified over their quadratic subfields K. The quadruplets L_{1}, L_{2}, L_{3}, L_{4} of nonisomorphic fields sharing the same discriminant among these totally real cubic fields are in onetoone correspondence with the 161 real quadratic fields K of 3class rank 2. Among the latter we have 149 with 3class group of type (3,3) and 12 with 3class group of type (9,3). This extensive database is the source of information for our third extension [6], which is presented here. 
Navigation: 0 < d < 100000 100000 < d < 200000 200000 < d < 300000 300000 < d < 400000 Reaching the Border of Ennola and Turunen's Domain: 400000 < d < 500000 Breaking Through Beyond Ennola and Turunen's Domain: 500000 < d < 600000 600000 < d < 700000 700000 < d < 800000 800000 < d < 900000 900000 < d < 1000000 

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