Remark.
For the statement of the theorems we must recall some concepts.
An unramified cyclic cubic relative extension N of a real quadratic field K
with relative automorphism group G = Gal(N|K) and unit group UN
is called of principal factorization type (or simply of type) Alpha,
if the cohomology group H0(G,UN) is non-trivial
(i. e., if the unit norm index (UK : NormN|KUN) equals 3)
and it is called of principal factorization type (or simply of type) Delta,
if the cohomology group H0(G,UN) is trivial
(i. e., if the unit norm index (UK : NormN|KUN) equals 1
and thus the fundamental unit of K is norm of a unit of N).
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Theorem 1. (for the proof see [1])
Let K be a real quadratic field with 3-class group of type (3,3).
Suppose that at least three of the four unramified cyclic cubic extensions N1,...,N4 of K
are of type Alpha, say N2,N3,N4.
Let the 3-class numbers of the non-Galois cubic subfields be
(h(L1),h(L2),h(L3),h(L4)) = (3u,3,3,3) with u ≥ 1.
Then the metabelian 3-group Gal(K2|K) of automorphisms of the second Hilbert 3-class field K2 of K
is of maximal class
and belongs to CF2a(2u+2) with symbolic order Y2u, if N1 is of type Delta,
and to CF2b(2u+2) with symbolic order Y2u'
and u ≥ 2, if N1 is of type Alpha.
The third possibility that Gal(K2|K)
belongs to CFa(2u+1) with symbolic order Y2u-1, if N1 is of type Alpha,
does obviously not occur (weak leaf conjecture).
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Theorem 2. (for the proof see [1])
Let K be either a real quadratic field with 3-class group of type (3,3)
such that all four unramified cyclic cubic extensions N1,...,N4 of K
are of type Delta,
or a complex quadratic field with 3-class group of type (3,3).
Let the 3-class numbers of the non-Galois cubic subfields be
(h(L1),h(L2),h(L3),h(L4)) = (3u,3v,3,3) with u ≥ v ≥ 1.
Then the metabelian 3-group Gal(K2|K) of automorphisms of the second Hilbert 3-class field K2 of K
is at most of second maximal class
and belongs either to CBFa(2u+2,2u+2v+1) with symbolic order R2u,2v
or to CBFb(2u+3,2u+2v+2).
The isomorphism invariant e has the odd value e = 2v+1.
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