1. The impact of number theoretic data on the structure of the group Gal(K_{2}K),
the group of automorphisms of the 2^{nd} Hilbert 3class field K_{2} over a quadratic field K
[1].

1.1. Theorems concerning the impact of 3class numbers.

1.2. Theorems concerning the impact of 3capitulation.

*

2. Real quadratic fields K with new capitulation types.

710652 is the smallest discriminant
[2]
of a real quadratic field K
with
capitulation type b.10 (0,0,4,3)
,
symbolic order V_{4,4}, and group G=Gal(K_{2}K) in CBF^{1b}(6,8)
of lower than second maximal class.
The 3class groups of N_{1} and N_{2} are of type (9,9),
but we have the exceptional type (3,3,3) for N_{3} and N_{4}.

631769 is the smallest discriminant
[2]
of a real quadratic field K
with
capitulation type D.5 (3,4,3,4)
,
symbolic order L_{2}, and group G=Gal(K_{2}K) in CBF^{1a}(4,5)
of second maximal class.
The 3class groups of N_{3} and N_{4} are of type (9,3),
but we have the exceptional type (3,3,3) for N_{1} and N_{2}.

540365 is the smallest discriminant
[2]
of a real quadratic field K
with
capitulation type c.21 (0,2,3,1)
,
symbolic order X_{3},
and nonterminal group G=Gal(K_{2}K) in CBF^{2a}(5,6)
of second maximal class.
Important remark:
This example shows that the strict leaf conjecture
that only terminal nodes (leaves) can occur as Gal(K_{2}K) is false.
However, it might still be true that a group cannot occur as Gal(K_{2}K),
if one of its successors is a leaf in CBF^{b} with the same capitulation type.
The 3class groups are of type (9,3) for N_{2}, N_{3}, and N_{4},
and of type (9,9) for N_{1}.

534824 is the smallest discriminant
[2]
of a real quadratic field K
with
capitulation type c.18 (0,3,1,3)
,
symbolic order X_{3},
and nonterminal group G=Gal(K_{2}K) in CBF^{2a}(5,6)
of second maximal class.
Important remark:
This example shows that the strict leaf conjecture
that only terminal nodes (leaves) can occur as Gal(K_{2}K) is false.
However, it might still be true that a group cannot occur as Gal(K_{2}K),
if one of its successors is a leaf in CBF^{b} with the same capitulation type.
The 3class groups are of type (9,3) for N_{2} and N_{4},
of type (9,9) for N_{1}
but we have the exceptional type (3,3,3) for N_{3}.

494236 is the smallest discriminant
[2]
of a real quadratic field K
with
an excited state of capitulation type a.3 (3,0,0,0)
,
symbolic order Y_{4}, and group G=Gal(K_{2}K) in CF^{2a}(6)
of maximal class.
The 3class groups are of type (3,3) for N_{2}, N_{3}, and N_{4},
and of type (27,9) for N_{1}.

422573 is the smallest discriminant
[2]
of a real quadratic field K
with
capitulation type D.10 (1,3,4,1)
,
symbolic order L_{2}, and group G=Gal(K_{2}K) in CBF^{1a}(4,5)
of second maximal class.
The 3class groups of N_{1}, N_{3}, and N_{4} are of type (9,3),
but we have the exceptional type (3,3,3) for N_{2}.

342664 is the smallest discriminant
[2]
of a real quadratic field K
with
capitulation type E.9 (4,1,3,4)
,
symbolic order X_{4}, and group G=Gal(K_{2}K) in CBF^{2a}(6,7)
of second maximal class.
The 3class groups are of type (9,3) for N_{2}, N_{3}, and N_{4},
and of type (27,9) for N_{1}.

214712 is the smallest discriminant
[2]
of a real quadratic field K
with
capitulation type G.19 (4,3,2,1)
,
symbolic order Z, and group G=Gal(K_{2}K) in CBF^{1b}(5,6)
of second maximal class.
Similarly as over complex quadratic fields,
in none of the four unramified cyclic cubic extensions NK
the complete 3class group of K becomes principal.
The 3class groups are of type (9,3) for N_{1}, N_{2}, N_{3}, and N_{4}.

*

3. Real quadratic fields K with finite 3class field tower K < K_{1} < K_{2} of length 2.
Since the
ground states of the capitulation types a.2 and a.3
are associated with
2stage metabelian 3groups G=Gal(K_{2}K) in CF^{2a}(4) of maximal class
(with commutator factor group G/G´ of type (3,3)),
whose main commutator has
[1]
the symbolic order Y_{2} = R_{2,1} = (X^{2},Y),
we have many
new examples of 3class field towers of length 2
.
For 26 of the 30 real quadratic fields with discriminant 0 < d < 3*10^{5} and 3class group of type (3,3)
the capitulation type is a.2 or a.3 in the ground state and the 3class field tower is finite of length 2.

*

4. Complex quadratic fields K with new capitulation types.

262744 is the smallest absolute discriminant
[2]
of a complex quadratic field K
with
an excited state of capitulation type E.14 (2,4,4,1)
,
symbolic order X_{6}, and group G=Gal(K_{2}K) in CBF^{2a}(8,9) of second maximal class.

159208 is the smallest absolute discriminant
[2]
of a complex quadratic field K
with
an excited state of capitulation type F.13 (2,3,4,3)
,
symbolic order R_{6,4}, and group G=Gal(K_{2}K) in CBF^{2a}(8,11) of lower than second maximal class.

124363 is the smallest absolute discriminant
[2]
of a complex quadratic field K
with the mysterious unknown (up to now)
capitulation type F.7 (4,4,1,1)
,
symbolic order R_{4,4}, and group G=Gal(K_{2}K) in CBF^{1a}(6,9) of lower than second maximal class.

21668 is the smallest absolute discriminant
[2]
of a complex quadratic field K
with
an excited state of capitulation type H.4 (2,1,1,1)
,
symbolic order Z_{5}, and group G=Gal(K_{2}K) in CBF^{2b}(7,8) of second maximal class.

