Our Mission:
1. To advance European science
to the forefront
of international research
and to stabilize this position.
2. To strengthen cooperation
with international research centers:

HarishChandra Research Institute,
Allahabad, India

Washington and Lee University,
Lexington, Virginia, USA

Université Mohammed Premier,
Oujda, Morocco

Australian National University,
Canberra, Capital Territory

Aichi University of Education,
Nagoya, Japan

University of Calgary,
Alberta, Canada
International Conferences:
February 27  29, 2020:
3rd Int. Conf. on Mathematics and its Applications
Université Hassan II, Faculté des Sciences
Casablanca, Region Occidental, Morocco
Daniel C. Mayer's Invited Key Note:
Abstract
Pattern Recognition via Artin Transfers
Program
April 25  27, 2019:
Conference on Algebra, Number Theory and Their Applications
Université Mohammed Premier, Faculté des Sciences
Oujda, Region Oriental, Morocco
Daniel C. Mayer's 3 Invited Lectures:
1. Proving the Conjecture of Scholz
2. Differential principal factors
3. Pure Metacyclic Fields
January 21  25, 2019:
AsiaAustralia Algebra Conference 2019
Western Sydney University, Parramatta City campus
Sydney, New South Wales, Australia
July 1  5, 2019:
31st Journées Arithmétiques
Istanbul University, Faculty of Science
JA 2019, Istanbul, Turkey

The aim of this succinct presentation:

Granting Worldwide Quick Access to Scientific Information

Reliable and Verifiable Contents by Citing PeerReviewed Papers as References
For quadratic number fields K with elementary 3class group Cl_{3}(K) of rank two,
the Transfer Kernel Type (TKT) κ, Transfer Target Type (TTT) τ,
Second 3Class Group G_{2}=Gal(F_{2}/K) and 3Tower Group G_{∞}=Gal(F_{∞}/K),
i.e. the essential invariants of the 3class field tower K ≤ F_{1} ≤ F_{2} ≤…≤ F_{∞},
have been determined in a series of recent publications between 2012 and 2018.
In the sequel, absolute values of fundamental discriminants d are minimal, and
finite 3groups are identified by their designation in the SmallGroups Database.
Imaginary Quadratic Fields, K=Q(d^{1/2}), d<0
d

TKT

κ

τ

G_{2}

G_{∞}

Ref.

3896

H.4

(4443)

[111,111,21,111]

⟨729,45⟩

⟨6561,606⟩

4.

4027

D.10

(2241)

[21,21,111,21]

⟨243,5⟩

⟨243,5⟩

6.

9748

E.9

(2231)

[32,21,21,21]

⟨2187,302⟩

⟨6561,620⟩

5.




or

⟨2187,306⟩

⟨6561,624⟩


12131

D.5

(4224)

[111,21,111,21]

⟨243,7⟩

⟨243,7⟩

6.

15544

E.6

(1313)

[32,21,111,21]

⟨2187,288⟩

⟨6561,616⟩

4.

16627

E.14

(2313)

[32,21,111,21]

⟨2187,289⟩

⟨6561,617⟩

4.




or

⟨2187,290⟩

⟨6561,618⟩


34867

E.8

(1231)

[32,21,21,21]

⟨2187,304⟩

⟨6561,622⟩

4.

Real Quadratic Fields, K=Q(d^{1/2}), d>0
d

TKT

κ

τ

G_{2}

G_{∞}

Ref.

32009

a.3

(2000)

[21,11,11,11]

⟨81,8⟩

⟨81,8⟩

6.

62501

a.1

(0000)

[22,11,11,11]

⟨729,99⟩

⟨729,99⟩

1.

72329

a.2

(1000)

[21,11,11,11]

⟨81,10⟩

⟨81,10⟩

6.

142097

a.3

(2000)

[111,11,11,11]

⟨81,7⟩

⟨81,7⟩

6.

152949

a.1

(0000)

[22,11,11,11]

⟨729,100⟩

⟨729,100⟩

1.

214712

G.19

(4321)

[21,21,21,21]

⟨729,57⟩

⟨2187,311⟩

2.

252977

a.1

(0000)

[22,11,11,11]

⟨729,101⟩

⟨729,101⟩

1.

342664

E.9

(2231)

[32,21,21,21]

⟨2187,302⟩

⟨6561,620⟩

2.




or

⟨2187,306⟩

⟨6561,624⟩


494236

a.3↑

(2000)

[32,11,11,11]

⟨729,97⟩

⟨729,97⟩

6.




or

⟨729,98⟩

⟨729,98⟩


534824

c.18

(0313)

[22,21,111,21]

⟨729,49⟩

⟨2187,291⟩

3.

540365

c.21

(0231)

[22,21,21,21]

⟨729,54⟩

⟨2187,307⟩

3.





or

⟨2187,308⟩


790085

a.2↑

(1000)

[32,11,11,11]

⟨729,96⟩

⟨729,96⟩

6.

957013

H.4

(4443)

[111,111,21,111]

⟨729,45⟩

⟨2187,273⟩

2.

2905160

a.1↑

(0000)

[33,11,11,11]

⟨6561,2227⟩

⟨6561,2227⟩

1.

3918837

E.14

(2313)

[32,21,111,21]

⟨2187,289⟩

⟨2187,289⟩

2.




or

⟨2187,290⟩

⟨2187,290⟩


4760877

E.9

(2231)

[32,21,21,21]

⟨2187,302⟩

⟨2187,302⟩

2.




or

⟨2187,306⟩

⟨2187,306⟩


5264069

E.6

(1313)

[32,21,111,21]

⟨2187,288⟩

⟨6561,616⟩

2.

6098360

E.8

(1231)

[32,21,21,21]

⟨2187,304⟩

⟨6561,622⟩

2.

7153097

E.6

(1313)

[32,21,111,21]

⟨2187,288⟩

⟨2187,288⟩

2.

8632716

E.8

(1231)

[32,21,21,21]

⟨2187,304⟩

⟨2187,304⟩

2.

9433849

E.14

(2313)

[32,21,111,21]

⟨2187,289⟩

⟨6561,617⟩

2.




or

⟨2187,290⟩

⟨6561,618⟩


10200108

a.3↑↑

(2000)

[43,11,11,11]

⟨6561,2223⟩

⟨6561,2223⟩

1.




or

⟨6561,2224⟩

⟨6561,2224⟩


10399596

a.1↑

(0000)

[33,11,11,11]

⟨6561,2225⟩

⟨6561,2225⟩

1.

14458876

a.2↑↑

(1000)

[43,11,11,11]

⟨6561,2222⟩

⟨6561,2222⟩

1.

27780297

a.1↑

(0000)

[33,11,11,11]

⟨6561,2226⟩

⟨6561,2226⟩

1.

References:

D. C. Mayer,
Deep transfers of pclass tower groups,
J. Appl. Math. Phys.
6
(2018),
no. 1,
3650,
DOI 10.4236/jamp.2018.61005.

D. C. Mayer,
Criteria for threestage towers of pclass fields,
Adv. Pure Math.
7
(2017),
no. 2,
135179,
DOI 10.4236/apm.2017.72008.

D. C. Mayer,
New number fields with known pclass tower,
Tatra Mt. Math. Pub.
64
(2015),
2157,
DOI 10.1515/tmmp20150040.

D. C. Mayer,
Indexp abelianization data of pclass tower groups,
Adv. Pure Math.
5
(2015)
no. 5,
286313,
DOI 10.4236/apm.2015.55029.

M. R. Bush and D. C. Mayer,
3class field towers of exact length 3,
J. Number Theory
147
(2015),
766777,
DOI 10.1016/j.jnt.2014.08.010.

D. C. Mayer,
Principalization algorithm via class group structure,
J. Théor. Nombres Bordeaux
26
(2014),
no. 2,
415464,
DOI 10.5802/jtnb.874.

D. C. Mayer,
The distribution of second pclass groups on coclass graphs,
J. Théor. Nombres Bordeaux
25
(2013),
no. 2,
401456,
DOI 10.5802/jtnb.842.

D. C. Mayer,
Transfers of metabelian pgroups,
Monatsh. Math.
166
(2012),
no. 34,
467495,
DOI 10.1007/s006050100277x.

D. C. Mayer,
The second pclass group of a number field,
Int. J. Number Theory
8
(2012),
no. 2,
471505,
DOI 10.1142/S179304211250025X.


Principal Investigator and
Project Leader of several
International Scientific Research Lines:
Supported by the
Austrian Science Fund (FWF)
J0497PHY and P 26008N25
and by the
Research Executive Agency of the
European Union (EUREA)
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