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.
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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.
State | m | M | G | G´ | (h(1),...,h(4)) | Minimal occurrence |
Ground state | 4 | Y2 | CF2a(4) | (3,3) | (3,3,3,3) | 32009 |
1st excited state | 6 | Y4 | CF2a(6) | (9,9) | (9,3,3,3) | 494236 |
2nd excited state | 8 | Y6 | CF2a(8) | (27,27) | (27,3,3,3) | unknown |
... | ... | ... | ... | ... | ... | ... |
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*
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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.
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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.
State | m | M | G | G´ | (h(1),...,h(4)) | Minimal occurrence |
Ground state | 5 | X3 | CBF2a(5,6) | (9,3,3) | (9,3,3,3) | 540365 |
1st excited state | 7 | X5 | CBF2a(7,8) | (27,9,3) | (27,3,3,3) | 1001957 |
2nd excited state | 9 | X7 | CBF2a(9,10) | (81,27,3) | (81,3,3,3) | unknown |
... | ... | ... | ... | ... | ... | ... |
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*
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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.
State | m | M | G | G´ | (h(1),...,h(4)) | Complex example | Real example |
Ground state | 6 | X4 | CBF2a(6,7) | (9,9,3) | (9,3,3,3) | -9748 | 342664 |
1st excited state | 8 | X6 | CBF2a(8,9) | (27,27,3) | (27,3,3,3) | -262744 | unknown |
2nd excited state | 10 | X8 | CBF2a(10,11) | (81,81,3) | (81,3,3,3) | unknown | unknown |
... | ... | ... | ... | ... | ... | ... | ... |
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