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Reliable and Verifiable Scientific Information on

3-CLASS FIELD TOWERS of QUADRATIC FIELDS



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with international research centers:
  • Harish-Chandra 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:
Asia-Australia 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



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The aim of this succinct presentation:
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  2. Reliable and Verifiable Contents by Citing Peer-Reviewed Papers as References



For quadratic number fields K with elementary 3-class group Cl3(K) of rank two,
the Transfer Kernel Type (TKT) κ, Transfer Target Type (TTT) τ,
Second 3-Class Group G2=Gal(F2/K) and 3-Tower Group G=Gal(F/K),
i.e. the essential invariants of the 3-class field tower K ≤ F1 ≤ F2 ≤…≤ 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 3-groups are identified by their designation in the SmallGroups Database.



Imaginary Quadratic Fields, K=Q(d1/2), d<0

d TKT κ τ G2 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(d1/2), d>0

d TKT κ τ G2 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:

  1. D. C. Mayer, Deep transfers of p-class tower groups,
    J. Appl. Math. Phys. 6 (2018), no. 1, 36-50, DOI 10.4236/jamp.2018.61005.

  2. D. C. Mayer, Criteria for three-stage towers of p-class fields,
    Adv. Pure Math. 7 (2017), no. 2, 135-179, DOI 10.4236/apm.2017.72008.

  3. D. C. Mayer, New number fields with known p-class tower,
    Tatra Mt. Math. Pub. 64 (2015), 21-57, DOI 10.1515/tmmp-2015-0040.

  4. D. C. Mayer, Index-p abelianization data of p-class tower groups,
    Adv. Pure Math. 5 (2015) no. 5, 286-313, DOI 10.4236/apm.2015.55029.

  5. M. R. Bush and D. C. Mayer, 3-class field towers of exact length 3,
    J. Number Theory 147 (2015), 766-777, DOI 10.1016/j.jnt.2014.08.010.

  6. D. C. Mayer, Principalization algorithm via class group structure,
    J. Théor. Nombres Bordeaux 26 (2014), no. 2, 415-464, DOI 10.5802/jtnb.874.

  7. D. C. Mayer, The distribution of second p-class groups on coclass graphs,
    J. Théor. Nombres Bordeaux 25 (2013), no. 2, 401-456, DOI 10.5802/jtnb.842.

  8. D. C. Mayer, Transfers of metabelian p-groups,
    Monatsh. Math. 166 (2012), no. 3-4, 467-495, DOI 10.1007/s00605-010-0277-x.

  9. D. C. Mayer, The second p-class group of a number field,
    Int. J. Number Theory 8 (2012), no. 2, 471-505, DOI 10.1142/S179304211250025X.



Daniel C. Mayer


Principal Investigator and
Project Leader of several

International Scientific Research Lines:

Supported by the
Austrian Science Fund (FWF)
J0497-PHY and P 26008-N25
and by the
Research Executive Agency of the
European Union (EUREA)


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