1. The impact of number theoretic data on the structure of the group Gal(K2|K),
the group of automorphisms of the 2nd Hilbert 3-class field K2 over a quadratic field K
[1].
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1.1. Theorems concerning the impact of 3-class numbers.
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1.2. Theorems concerning the impact of 3-capitulation.
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2. Real quadratic fields K with new capitulation types.
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710652 is the smallest discriminant
[2]
of a real quadratic field K
with
capitulation type b.10 (0,0,4,3)
,
symbolic order V4,4, and group G=Gal(K2|K) in CBF1b(6,8)
of lower than second maximal class.
The 3-class groups of N1 and N2 are of type (9,9),
but we have the exceptional type (3,3,3) for N3 and N4.
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631769 is the smallest discriminant
[2]
of a real quadratic field K
with
capitulation type D.5 (3,4,3,4)
,
symbolic order L2, and group G=Gal(K2|K) in CBF1a(4,5)
of second maximal class.
The 3-class groups of N3 and N4 are of type (9,3),
but we have the exceptional type (3,3,3) for N1 and N2.
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540365 is the smallest discriminant
[2]
of a real quadratic field K
with
capitulation type c.21 (0,2,3,1)
,
symbolic order X3,
and non-terminal group G=Gal(K2|K) in CBF2a(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(K2|K) is false.
However, it might still be true that a group cannot occur as Gal(K2|K),
if one of its successors is a leaf in CBFb with the same capitulation type.
The 3-class groups are of type (9,3) for N2, N3, and N4,
and of type (9,9) for N1.
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534824 is the smallest discriminant
[2]
of a real quadratic field K
with
capitulation type c.18 (0,3,1,3)
,
symbolic order X3,
and non-terminal group G=Gal(K2|K) in CBF2a(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(K2|K) is false.
However, it might still be true that a group cannot occur as Gal(K2|K),
if one of its successors is a leaf in CBFb with the same capitulation type.
The 3-class groups are of type (9,3) for N2 and N4,
of type (9,9) for N1
but we have the exceptional type (3,3,3) for N3.
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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 Y4, and group G=Gal(K2|K) in CF2a(6)
of maximal class.
The 3-class groups are of type (3,3) for N2, N3, and N4,
and of type (27,9) for N1.
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422573 is the smallest discriminant
[2]
of a real quadratic field K
with
capitulation type D.10 (1,3,4,1)
,
symbolic order L2, and group G=Gal(K2|K) in CBF1a(4,5)
of second maximal class.
The 3-class groups of N1, N3, and N4 are of type (9,3),
but we have the exceptional type (3,3,3) for N2.
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342664 is the smallest discriminant
[2]
of a real quadratic field K
with
capitulation type E.9 (4,1,3,4)
,
symbolic order X4, and group G=Gal(K2|K) in CBF2a(6,7)
of second maximal class.
The 3-class groups are of type (9,3) for N2, N3, and N4,
and of type (27,9) for N1.
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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(K2|K) in CBF1b(5,6)
of second maximal class.
Similarly as over complex quadratic fields,
in none of the four unramified cyclic cubic extensions N|K
the complete 3-class group of K becomes principal.
The 3-class groups are of type (9,3) for N1, N2, N3, and N4.
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3. Real quadratic fields K with finite 3-class field tower K < K1 < K2 of length 2.
Since the
ground states of the capitulation types a.2 and a.3
are associated with
2-stage metabelian 3-groups G=Gal(K2|K) in CF2a(4) of maximal class
(with commutator factor group G/G´ of type (3,3)),
whose main commutator has
[1]
the symbolic order Y2 = R2,1 = (X2,Y),
we have many
new examples of 3-class field towers of length 2
.
For 26 of the 30 real quadratic fields with discriminant 0 < d < 3*105 and 3-class group of type (3,3)
the capitulation type is a.2 or a.3 in the ground state and the 3-class field tower is finite of length 2.
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4. Complex quadratic fields K with new capitulation types.
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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 X6, and group G=Gal(K2|K) in CBF2a(8,9) of second maximal class.
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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 R6,4, and group G=Gal(K2|K) in CBF2a(8,11) of lower than second maximal class.
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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 R4,4, and group G=Gal(K2|K) in CBF1a(6,9) of lower than second maximal class.
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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 Z5, and group G=Gal(K2|K) in CBF2b(7,8) of second maximal class.
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