
Original Problem.
In the case of an odd prime p,
algebraic number fields K with
Hilbert pclass field tower
K = F_{p}^{0}(K) <
F_{p}^{1}(K) <
F_{p}^{2}(K) < … <
F_{p}^{n}(K) =
F_{p}^{∞}(K)
of finite length L_{p}(K) = n bigger than two
were
unknown
until August 24, 2012.
This day was the historic event of the discovery of
complex quadratic fields K = Q(D^{1/2})
with 3class field towers consisting of exactly three stages,
that is L_{3}(K) = 3.

Modified Problem.
All these fields had 3class groups of type (3,3)
and it remained an open problem whether there exist
number fields K with nonelementary pclass group,
and finite length L_{p}(K) > 2,
for an odd prime p.
It was the 5^{th} of May, 2013,
that we succeeded in providing evidence for the existence of
3class field towers of exactly three stages
over complex quadratic fields with
3class groups of nonelementary type (3,9).

Presentation of Recent Results
of May 5, 2013.
Let D denote the discriminant
of a complex quadratic field K.
We begin with arithmetical data whose investigation
goes back to
2009 and 2010
,
and was completed at the end of
July 2011
.
Punctured transfer kernel types (TKT)
and transfer target types (TTT)
were defined in a
preprint 2011
.

For D = 42567, the field K has
punctured TKT C.4, (3,1,1;3),
and homocyclic TTT [(9,9,9),(3,27),(3,27),(3,3,9)].

For D = 35331, the field K has
punctured TKT C.4, (3,1,1;3),
and heterocyclic TTT [(3,9,27),(3,27),(3,27),(3,3,9)].

For D = 116419, the field K has
punctured TKT D.5, (2,1,1;3),
and homocyclic TTT [(9,9,9),(3,27),(3,27),(3,3,9)].

For D = 17723, the field K has
punctured TKT D.5, (2,1,1;3),
and heterocyclic TTT [(3,9,27),(3,27),(3,27),(3,3,9)].

For D = 11651, the field K has
punctured TKT D.10, (4,1,1;3),
and TTT [(3,9,27),(3,27),(3,27),(3,3,9)].

For D = 31983, the field K has
punctured TKT D.6, (1,3,2;1),
and TTT [(3,27),(3,27),(3,27),(3,9,27)].
Now we present our new theorem,
which links algebraic number theory with pgroup theory via class field theory.
In the sequel,
we use the notation of the
SmallGroups library
and
ANUPQ package
of
GAP
and
MAGMA
for identifying pgroups and their descendants.
Theorem. (Mayer)
Let K be a complex quadratic field
with 3class group of type (3,9) and
denote by G the Galois group
Gal(F_{3}^{∞}(K)K)
of the maximal unramified pro3 extension of K.

If K has
punctured TKT C.4, (3,1,1;3),
and homocyclic TTT [(9,9,9),(3,27),(3,27),(3,3,9)],
then its 3tower group G is determined uniquely as
<2187,168>#2;2,
a Schur σgroup of derived length 3, order 19683 and coclass 4,
having 2^{nd} derived quotient
G/G'' ≅ <2187,168>#1;2
of order 6561 and coclass 3.

If K has
punctured TKT C.4, (3,1,1;3),
and heterocyclic TTT [(3,9,27),(3,27),(3,27),(3,3,9)],
then its 3tower group G is determined uniquely as
<2187,168>#2;6,
a Schur σgroup of derived length 3, order 19683 and coclass 4,
having metabelianization
G/G'' ≅ <2187,168>#1;6
of order 6561 and coclass 3.

If K has
punctured TKT D.5, (2,1,1;3),
and homocyclic TTT [(9,9,9),(3,27),(3,27),(3,3,9)],
then its 3tower group G is determined uniquely as
<2187,168>#2;3,
a Schur σgroup of derived length 3, order 19683 and coclass 4,
having metabelianization
G/G'' ≅ <2187,168>#1;3
of order 6561 and coclass 3.

If K has
punctured TKT D.5, (2,1,1;3),
and heterocyclic TTT [(3,9,27),(3,27),(3,27),(3,3,9)],
then its 3tower group G is determined uniquely as
<2187,168>#2;5,
a Schur σgroup of derived length 3, order 19683 and coclass 4,
having 2^{nd} derived quotient
G/G'' ≅ <2187,168>#1;5
of order 6561 and coclass 3.

If K has
punctured TKT D.10, (4,1,1;3),
and TTT [(3,9,27),(3,27),(3,27),(3,3,9)],
then its 3tower group G is
either <2187,168>#2;8 or <2187,168>#2;9.
Both are Schur σgroups of derived length 3, order 19683 and coclass 4,
having metabelianizations
G/G'' ≅ <2187,168>#1;8 resp. <2187,168>#1;9.

If K has
punctured TKT D.6, (1,3,2;1),
and TTT [(3,27),(3,27),(3,27),(3,9,27)],
then its 3tower group G is
either <2187,181>#2;4 or <2187,191>#2;4.
Both are Schur σgroups of derived length 3, order 19683 and coclass 4,
having 2^{nd} derived quotients
G/G'' ≅ <2187,181>#1;4 resp. <2187,191>#1;4.
The proof is a consequence of the detailed table
in § 1.2 of our
data collection about descendant trees and Schur σgroups
.
In fact, the table even determines the coclass tree and branch
where G/G'' is located.

Recall of Joint Results
of August 24, 2012.
To emphasize the similar structure of both theorems
of May 5, 2013, and of August 24, 2012,
we express the latter by means of the same concepts.
Theorem. (Boston, Bush, Mayer)
Let K be a complex quadratic field
with 3class group of type (3,3) and
denote by G the Galois group
Gal(F_{3}^{∞}(K)K)
of the maximal unramified pro3 extension of K.

If K has
TKT E.6, (1,1,2,2),
and TTT [(9,27),(3,3,3),(3,9),(3,9)],
then its 3tower group G is determined uniquely as
<729,49>#2;4,
a Schur σgroup of derived length 3, order 6561 and coclass 3,
having 2^{nd} derived quotient
G/G'' ≅ <729,49>#1;5 = <2187,288>
of coclass 2.

If K has
TKT E.14, (3,1,2,2),
and TTT [(9,27),(3,3,3),(3,9),(3,9)],
then its 3tower group G is
either <729,49>#2;5 or <729,49>#2;6.
Both are Schur σgroups of derived length 3, order 6561 and coclass 3,
having metabelianizations
G/G'' ≅ <729,49>#1;6 = <2187,289>
resp. <729,49>#1;7 = <2187,290>
of coclass 2.

If K has
TKT E.8, (2,2,3,4),
and TTT [(3,9),(9,27),(3,9),(3,9)],
then its 3tower group G is determined uniquely as
<729,54>#2;4,
a Schur σgroup of derived length 3, order 6561 and coclass 3,
having metabelianization
G/G'' ≅ <729,54>#1;4 = <2187,304>
of coclass 2.

If K has
TKT E.9, (2,3,3,4),
and TTT [(3,9),(9,27),(3,9),(3,9)],
then its 3tower group G is
either <729,54>#2;2 or <729,54>#2;6.
Both are Schur σgroups of derived length 3, order 6561 and coclass 3,
having 2^{nd} derived quotients
G/G'' ≅ <729,54>#1;2 = <2187,302>
resp. <729,54>#1;6 = <2187,306>
of coclass 2.
The proof is a consequence of the detailed table
in § 1.1 of our
data collection about descendant trees and Schur σgroups
.
In fact, the table even determines the coclass tree and branch
where G/G'' is located:
branch B(6) of tree T(<243,6>) for the immediate descendants of <729,49>, and
branch B(6) of tree T(<243,8>) for the immediate descendants of <729,54>.
Let us illustrate the BostonBushMayer Theorem by arithmetical data whose
investigation goes back to
2003 and 2004
,
partially even to
1933 and 1982
.
Transfer kernel types (TKT)
and transfer target types (TTT)
were defined in the
related presentation
.

For D = 15544, the field K has
TKT E.6, (1,1,2,2),
and TTT [(9,27),(3,3,3),(3,9),(3,9)].
(TKT by Heider and Schmithals, 1982)

For D = 16627, the field K has
TKT E.14, (3,1,2,2),
and TTT [(9,27),(3,3,3),(3,9),(3,9)].
(TKT by Heider and Schmithals, 1982)

For D = 34867, the field K has
TKT E.8, (2,2,3,4),
and TTT [(3,9),(9,27),(3,9),(3,9)].
(Mayer, 2003)
This is the unique capitulation type with 3 fixed points.
The discriminant D lies below the limit 20000 of Angell's 1972 table,
which was used by
Heider and Schmithals
in 1982.

For D = 9748, the field K has
TKT E.9, (2,3,3,4),
and TTT [(3,9),(9,27),(3,9),(3,9)].
(Scholz and Taussky, 1933)
This is the horribly obstinate field which caused 80 years of intolerable uncertainty,
since
Scholz and Taussky
erroneously claimed that it has a 3class field tower
of length two.
Nobody
was able to confirm or refute this assertion.
On August 23, 2012, Bush and Mayer disproved Scholz and Taussky's claim,
using the concept of Schur σgroups and providing evidence
for a tower of length at least three.
One day after, Boston, Bush and Mayer were even able
to prove that the tower consists of exactly three stages.

Important Alert.
We point out that there are two kinds of descendant trees
with respect to the Schur σcover of their metabelian vertices,
that is, the set of all nonmetabelian Schur σgroups
having a given vertex as their common metabelianization.

Descendant trees with finite Schur σcovers,
in fact, containing a unique element, in all known examples.
These trees consist of vertices with different TKT and
contain a complete coclass tree with metabelian mainline.
Examples are the trees with roots
<729,49> (TKT c.18), <729,54> (TKT c.21)
or <2187,168> (TKT d.10), <2187,181>, <2187,191> (both TKT e.14).

Descendant trees with infinite Schur σcovers.
All vertices of these trees share a common TKT and
only finitely many descendants are of the same coclass as the root.
Examples are the trees with roots
<729,45> (TKT H.4), <729,57> (TKT G.19)
or <2187,173>, <2187,183> (both TKT B.7), <2187,178>, <2187,188> (both TKT E.12).
For instance, <729,45>#2;2 is only one of
infinitely many
Schur σgroups G having the same metabelianization G/G'' ≅ <729,45>.




