# Theorems concerning the impact of 3-capitulation.

Theorem a. (for the proof see [1])
A real quadratic field K with 3-class group of type (3,3)
and capitulation type a.2 (1,0,0,0) or a.3 (0,1,0,0) determines
the metabelian 3-group G = Gal(K2|K) of automorphisms of the second Hilbert 3-class field K2 of K
and the symbolic order M of the main commutator [y,x] of G = <x,y> up to a single infinite degree of freedom as
G = G0m(1,0) in CF2a(m), for capitulation type a.2,
G = G0m(0,+-1) in CF2a(m), for capitulation type a.3,
with even m ≥ 4, and M = Ym-2.
The following table visualizes the begin of this series with a single infinite degree of freedom,
listing the index m of nilpotency, the symbolic order M, the group G, the commutator subgroup G´,
and the 3-class numbers of the four associated non-Galois cubic fields.
StatemMG(h(1),...,h(4))Minimal occurrence
Ground state4Y2CF2a(4)(3,3)(3,3,3,3)32009
1st excited state6Y4CF2a(6)(9,9)(9,3,3,3)494236
2nd excited state8Y6CF2a(8)(27,27)(27,3,3,3)unknown
.....................
*
Theorem c. (for the proof see [1])
A real quadratic field K with 3-class group of type (3,3)
and capitulation type c.18 (2,0,1,1) or c.21 (1,2,0,3) determines
the metabelian 3-group G = Gal(K2|K) of automorphisms of the second Hilbert 3-class field K2 of K
and the symbolic order M of the main commutator [y,x] of G = <x,y> up to a single infinite degree of freedom as
G = G0m,m+1(0,-1,0,1) in CBF2a(m,m+1), for capitulation type c.18,
G = G0m,m+1(0,0,0,1) in CBF2a(m,m+1), for capitulation type c.21,
with odd m ≥ 5, and M = Xm-2.
The following table visualizes the begin of this series with a single infinite degree of freedom,
listing the index m of nilpotency, the symbolic order M, the group G, the commutator subgroup G´,
and the 3-class numbers of the four associated non-Galois cubic fields.
StatemMG(h(1),...,h(4))Minimal occurrence
Ground state5X3CBF2a(5,6)(9,3,3)(9,3,3,3)540365
1st excited state7X5CBF2a(7,8)(27,9,3)(27,3,3,3)1001957
2nd excited state9X7CBF2a(9,10)(81,27,3)(81,3,3,3)unknown
.....................
*
Theorem E. (for the proof see [1])
A complex or real quadratic field K with 3-class group of type (3,3)
and capitulation type E.6 (1,1,2,2) or E.8 (1,2,3,1)
or E.9 (1,2,1,3) or E.14 (2,3,1,1) determines
the metabelian 3-group G = Gal(K2|K) of automorphisms of the second Hilbert 3-class field K2 of K
and the symbolic order M of the main commutator [y,x] of G = <x,y> up to a single infinite degree of freedom as
G = G0m,m+1(1,-1,1,1) in CBF2a(m,m+1), for capitulation type E.6,
G = G0m,m+1(1,0,-1,1) in CBF2a(m,m+1), for capitulation type E.8,
G = G0m,m+1(0,0,+-1,1) in CBF2a(m,m+1), for capitulation type E.9,
G = G0m,m+1(0,-1,+-1,1) in CBF2a(m,m+1), for capitulation type E.14,
with even m ≥ 6, and M = Xm-2.
The following table visualizes the begin of this series with a single infinite degree of freedom,
listing the index m of nilpotency, the symbolic order M, the group G, the commutator subgroup G´,
and the 3-class numbers of the four associated non-Galois cubic fields.
StatemMG(h(1),...,h(4))Complex exampleReal example
Ground state6X4CBF2a(6,7)(9,9,3)(9,3,3,3)-9748342664
1st excited state8X6CBF2a(8,9)(27,27,3)(27,3,3,3)-262744unknown
2nd excited state10X8CBF2a(10,11)(81,81,3)(81,3,3,3)unknownunknown
........................

 References: [1] Daniel C. Mayer, Two-Stage Towers of 3-Class Fields over Quadratic Fields, (Latest Update) Univ. Graz, 2008.

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