Reliable and Verifiable Scientific Information on

3-CLASS FIELD TOWERS of QUADRATIC FIELDS

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: 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 The aim of this succinct presentation: Granting Worldwide Quick Access to Scientific Information 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: 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. 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. 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. 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. 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. D. C. Mayer, Principalization algorithm via class group structure, J. Théor. Nombres Bordeaux 26 (2014), no. 2, 415-464, DOI 10.5802/jtnb.874. 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. D. C. Mayer, Transfers of metabelian p-groups, Monatsh. Math. 166 (2012), no. 3-4, 467-495, DOI 10.1007/s00605-010-0277-x. 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. Principal Investigator and Project Leader of several International Scientific Research Lines: Towers of p-Class Fields over Algebraic Number Fields Structured Descendant Trees of Finite p-Groups Lie Groups in General Relativity, Astrophysics, and Particle Physics Supported by the Austrian Science Fund (FWF) J0497-PHY and P 26008-N25 and by the Research Executive Agency of the European Union (EUREA) Our Services to the Mathematical Community: Mathematical Reviews (MR) Data Base