Real quadratic fields with 3class group of type (3,3).
Like all quadratic fields K of 3class rank 2,
they have 4 unramified cyclic cubic extensions N_{1},...,N_{4}
whose nonGalois cubic subfields L_{1},...,L_{4}
share the same (fundamental) discriminant as the quadratic field K.

Before 1982, it was unknown,
if the complete 3class group of K can become principal
in any of the extensions N_{i}, provided that K is real.
The numerical investigations [1] of F.P. Heider and B. Schmithals in 1982
showed that this phenomenon occurs indeed for the 5 smallest discriminants
1. 32009
2. 42817
3. 62501
4. 72329
5. 94636
These were the first examples of principal factorization type Alpha_{1}.
For d = 62501, the quadruplet of PFT's appears in pure form (A,A,A,A),
for the other four discriminants in mixed form (A,A,A,D)
with a single case of principal factorization type Delta_{1}
(having only a 1dimensional capitulation of ideal classes).

In the time from January 15^{th}, 2006, to February 6^{th}, 2006,
we investigated the sequence of these discriminants further [4] and found that
6. 103809
7. 114889
8. 130397
9. 142097
10. 151141
11. 152949
12. 153949
13. 172252
14. 173944
15. 184137
16. 189237
17. 206776
18. 209765
19. 213913
20. 214028
behave completely like the first 5 smallest discriminants. We have (known from [3] already)
pure PFT (A,A,A,A) for d = 152949, and mixed PFT's (A,A,A,D) otherwise.

However, on January 28^{th}, 2006,
we were extremely surprised when we discovered [4]
that all four totally real cubic fields of discriminant
21. 214712
are of principal factorization type Delta_{1}.
Of course we were very curious about the capitulation type
and on January 30^{th}, 2006, we determined it to be G.19, (4321).

On April 28^{th}, 2006, we found further occurrences of
quadruplets (D,D,D,D) with principal factorization type Delta_{1} for
37. 342664
45. 422573
and on June 13^{th}, 2006, we calculated their capitulation types as
E.9, (4134) and D.10, (1341), respectively.
Thus we have solved the last open questions for unramified extensions in
Ennola and Turunen's range 0 < d < 500000 of totally real cubic discriminants [2]
and we are pushing forward to the mysterious region 500000 < d < 1000000.

