2016

Nov 11:

International Colloquium of Algebra, Number Theory, Cryptography and Information Security

ANCI, Taza

Invited Keynote
on Recent Progress in Determining pClass Field Towers, Taza, Morocco

Sep 08:

Sensational Discovery of the First ThreeStage Tower

F_{3}^{∞}K/K

of 3Class Fields over a Cyclic Cubic Field K with Conductor 28791=9*7*457

Sep 07:

3Principalization Type and Second 3Class Group

Gal(F_{3}^{2}K/K)

of all Cyclic Cubic Fields K with 3Class Group (3,3) and Conductor c < 100000

Aug 20:

Long desired discovery of a Second 5Class Group of Elevated Coclass

cc(G_{5}^{2}M) ≥ 3

for the Cyclic Quartic 5Mirror Field M=Q((ζζ^{1})d^{1/2}) with d = 8689

Aug 09:

5Principalization Type and Second 5Class Group

Gal(F_{5}^{2}M/M)

of all Cyclic Quartic 5Mirror Fields M=Q((ζζ^{1})d^{1/2}), d < 10000, with 5Class Group (5,5)

Jul 25:

2nd International Conference on Groups and Algebras

ICGA, Suzhou

Invited Lecture
on pCapitulation over Number Fields with pClass Rank Two, Suzhou, China

Jun 28:

18th Postgraduate Group Theory Conference

Imperial College, London

Presentation ThreeStage Towers of 5Class Fields, London, UK

Jun 25:

Fork Topology, Contestants, and Shafarevich Covers

cov(G,K)

of metabelian 3groups G_{3}^{2}K with Kernel Type F and Coclass 6 over quadratic fields K

Jun 15:

Annihilator Ideals of TwoGenerated Metabelian pGroups

Article

establishing the general templates S,W,Y and verifying the specializations L,X,Z,R,T,V

Jun 07:

Precise Artin Patterns (kernels and targets of Artin transfers)

AP=(TKT,TTT)

of all 24476 complex quadratic fields with discriminant 10^{7} < d, by far most extensive computation up to now

May 30:

Recent Progress in Determining pClass Field Towers

Invited Lecture

for the International Colloquium on Algebra, Number Theory, Cryptography and Information Security in Taza, Morocco, 2016

Apr 29:

Sensational first realization of the 5Sylow subgroup of the symmetric group of degree 25 as

2nd 5class group

Occurrence for the real quadratic field with discriminant d = 27186289

Apr 07:

Sensational discovery of the first 5CLASS TOWER OF EXACT LENGTH 3

Priority Claim

Occurrence for the real quadratic fields with discriminants d = 3812377 and 19621905

Mar 16:

Precise Artin Patterns (kernels and targets of Artin transfers)

AP=(TKT,TTT)

of all 34631 real quadratic fields with discriminant d < 10^{8}, by far most extensive computation up to now

Mar 13:

Recent Progress in Determining pClass Field Towers

Abstract

for the International Colloquium on Algebra, Number Theory, Cryptography and Information Security in Taza, Morocco, 2016

Feb 28:

On class groups of imaginary quadratic fields

Math. Rev. 3404031

of a paper by A. Wiles, in J. London Math. Soc.

Feb 18:

Annual Report 2015 for the Austrian Science Fund (FWF)

Report

Standalone research project P 26008N25

2015

Nov 24:

Artin transfer patterns on descendant trees of finite pgroups

Adv. Pure Math.

6 (2016), no. 1, Special Issue on Group Theory Research

Nov 12:

GrunwaldWang theorem, an effective version

Math. Rev. 3368167

on a paper by S. Wang, in Sci. China Math.

Oct 18:

Induced homomorphism (quotient group)

Article

induced homomorphism, characteristic subgroups, GIautomorphisms, functorial properties

Sep 30:

New number fields with known pclass tower

Tatra Mountains Math. Publ.

64 (2015), 2157

Aug 31:

22nd Czech and Slovak International Conference on Number Theory (CSICNT) 2015, Liptovsky Jan, Slovakia

Presentation

New number fields with known pclass tower

Jul 21:

1st International Conference on Groups and Algebras (ICGA) 2015, Shanghai, China

Presentation

Invited Lecture on Periodic sequences of pclass tower groups

Jul 09:

29th Journées Arithmétiques (JA) 2015, Debrecen, Hungary

Presentation

Indexp abelianization data of pclass tower groups

May 18:

École de Recherche CIMPA UNESCO 2015

Oujda, Morocco

La théorie algorithmique des nombres (lectures and exercises)

Mar 25:

Higher Newton polygons and integral bases

Math. Rev. 3276340

of a paper by J. Guàrdia, J. Montes and E. Nart, in J. Number Theory

Mar 08:

Descendants of <64,75> as second 2class groups of biquadratic Dirichlet fields

Q((1)^{1/2},d^{1/2})

Special instances of the cases A and D by A. Azizi and M. Taous

Feb 25:

Annual Report 2014 for the Austrian Science Fund (FWF)

Report

Standalone research project P 26008N25

Feb 18:

Indexp abelianization data of pclass tower groups

Adv. Pure Math.

Special Issue on Number Theory and Cryptography

Jan 30:

Coclass of Gal(k_{2}^{(2)}k) for some fields k = Q((p_{1}p_{2}q)^{1/2},(1)^{1/2}) with 2class group of type (2,2,2)

Report

on a paper by A. Azizi, A. Zekhnini, and M. Taous, in Journal of Algebra and its Applications

Jan 27:

Periodic bifurcations in descendant trees of finite pgroups

Adv. Pure Math.

Special Issue on Group Theory

2014

Dec 10:

The group Gal(k_{3}^{(2)}k) for k = Q((3)^{1/2},d^{1/2}) of type (3,3)

IJNT

in cooperation with M. Talbi, A. Derhem, Mm. Talbi and A. Azizi

Dec 08:

Principalization of 2class groups of type (2,2,2) of biquadratic fields
Q((p_{1}p_{2}q)^{1/2},(1)^{1/2})

World Scientific

Int. J. Number Theory 11 (2015), no. 4, 11771215, DOI 10.1142/S1793042115500645,
in cooperation with A. Azizi, A. Zekhnini and M. Taous.

Nov 13:

Quadratic pring spaces for counting dihedral fields

World Scientific

Int. J. Number Theory 10 (2014), no. 8, 2205  2242, DOI 10.1142/S1793042114500754.

Oct 23:

Principalization algorithm via class group structure

CEDRAM

J. Théor. Nombres Bordeaux 26 (2014), no. 2, 415  464.

Oct 07:

Signed fundamental domains for totally real number fields

Math. Rev. 3198753

of a paper by F. Diaz y Diaz and E. Friedman, in Proc. London Math. Soc.

Sep 19:

Endowing descendant trees T(R), R = <81,2> and <81,5>, with structure

by implementation of the

Artin transfers for pgroups G having G/G' = (p^{2},p^{2}) or (p^{3},p)

Sep 18:

Entering uncharted waters by construction of descendant trees T(R) with

roots R = <81,2>, <81,5>,

consisting of 3groups G with abelianization G/G' = (9,9) or (27,3)

Sep 16:

Completely new perspective of the population of coclass trees T_{2}(R) with

roots R = <243,6>, <243,8>,

based on computations of IPADs by M. R. Bush and TKTs by D. C. Mayer

Aug 27:

Structural interpretation of composite Artin transfers

Article

Wreath product of symmetric groups S_{m} and S_{n} with respect to {1,…,n}

Jul 31:

pgroup generation algorithm

Article

pcovering group, nucleus, allowable subgroups, orbits, step sizes

Jul 19:

Complex quadratic fields of type (3,3,3)

Preprint

Infinite 3class tower, transfer kernels and targets, and second 3class group

Jul 12:

On the Hilbert 2class field tower of some abelian 2extensions of Q

Math. Rev. 3165518

of a paper by A. Azizi and A. Mouhib, in Czechoslovak Math. J.

Jul 05:

Principalization in algebraic number theory

Article

Artin reciprocity law, commutative Artin diagram, and Galois cohomology

Jun 16:

Artin transfers and structured descendant trees

Article

Inheritance from quotients, stabilization criteria, and pattern recognition

May 28:

Strategy of pattern recognition via Artin transfers

Preprint

Arithmetical applications of: commutative Artin diagram, inheritance from quotients

May 21:

Descendant trees of finite pgroups

Article

Nuclear rank, multifurcation, coclass graphs, parametrized pcpresentations

Apr 12:

Principalization of 2class groups of type (2,2,2) of biquadratic fields

k = Q((1)^{1/2},d^{1/2}), d = p_{1}p_{2}q

in cooperation with A. Azizi, A. Zekhnini, and M. Taous, in IJNT

Mar 27:

Annual Report 2013 for the Austrian Science Fund (FWF)

Report

Standalone research project P 26008N25

Feb 28:

A new computational approach to ideal theory in number fields

Math. Rev. 3105943

of a paper by J. Guàrdia, J. Montes, and E. Nart, in Found. Comput. Math.

Jan 31:

On the strongly ambiguous classes of kQ(i) where k = Q((2p_{1}p_{2})^{1/2},i)

Review

of a paper by A. Azizi, A. Zekhnini, and M. Taous, in AsianEuropean Journal of Mathematics

2013

Dec 18:

West Coast Number Theory 2013, Asilomar, Monterey, California

Presentation

Class towers and capitulation over quadratic fields

Nov 17:

Classification into principal factorization types of pure quintic fields

Q(D^{1/5})

by means of metacyclic class number relations and relative norms of fundamental units

Nov 06:

Using Pari/GP and Magma for the classification of pure cubic fields

Q(D^{1/3})

into principal factorization types, according to Barrucand / Cohn

Nov 03:

Using ANUPQ and Magma for descendant generation of <729,9…12>

of type (9,3)

as an important supplement to the descendant trees of <729,13…21>

Oct 25:

Units of pure quartic fields Q(p^{1/4}) with p ≡ 7 (mod 16)

Math. Rev. 3059109

of a paper by A. AguilarZavoznik and M. PinedaRuelas, in Far East J. Math. Sci.

Oct 20:

Determining the index of relative units of pure quartic fields

Q(D^{1/4})

by means of calculating the relative norms of a fundamental system of units

Oct 07:

Using ANUPQ and Magma for descendant generation of <243,3>

of type (3,3)

to get the complete overview of transfer kernel types in section F

Sep 23:

ÖMG and DMV Congress 2013, Innsbruck, Austria

Presentation

Abstract
,
3class field towers of exact length 3

Sep 01:

Official Start of our International Research Project

FWF: P 26008N25

Towers of pClass Fields over Algebraic Number Fields

Aug 30:

French translation of the CIMPA  UNESCO Lectures "CIMPA 2015 Oujda"

Resumé

Théorie des Nombres Algorithmique et Applications en Cryptographie

Aug 25:

Complete Proposal for the CIMPA  UNESCO Course "CIMPA 2015 Oujda"

Summary

Computational Number Theory and its Applications in Cryptography

Aug 22:

Inspired by some presentations at "Groups St Andrews 2013":

Preprint

Complete Normal Lattice of metabelian pgroups G, G/G' ≅ (p,p), k(G) = 0

Aug 19:

Generalization of our lecture at "Groups Saint Andrews 2013"

Preprint

PCPresentations for infinite periodic sequences of 3groups G, G/G' ≅ (9,3)

Aug 03:

Groups St. Andrews 2013, Fife, Scotland

Presentation

Abstract
,
Finite 3groups as viewed from class field theory

Jul 05:

Celebrating the Thirtieth Anniversary of my Ph. D. Promotion

July 5th, 1983

Periodicity and Structure of NonMetabelian Relators, Zassenhaus Filtration

Jul 03:

Nonmetabelian finite 3groups G with transfer kernel type κ(G) in

Section E, cc(G) = 3

First NonMetabelian Parametrized PCPresentations

Jul 02:

First step to find all finite 3groups G with transfer kernel type κ(G) in

Section E, cc(G) = 2

New Metabelian Parametrized PCPresentations

Jun 17:

The distribution of second pclass groups on coclass graphs

CEDRAM

J. Théor. Nombres Bordeaux 25 (2013), no. 2, 401  456.

Jun 09:

Tough computation of second layer pTKTs κ_{2} over complex quadratic fields

of type (3,9), such as

K = Q((64671)^{1/2}) of pTKT A.20, κ = (444;4), having a sensational κ_{2} = (333;2)

May 28:

Discovery of further 3towers having three stages over number fields

of type (3,9), such as

K = Q((320968)^{1/2}) of pTKT C.4 in its 1^{st} excited state, by D. C. Mayer

May 05:

First evidence of 3class field towers having three stages over number fields

of type (3,9),

for example K = Q((42567)^{1/2}) of pTKT C.4, κ = (112;2), by D. C. Mayer

Apr 13:

Criteria for sextic Pólya fields of type K = Q(ζ,D^{1/3}),

ζ = exp(2πi/3),

based on investigations by Amandine Leriche (Amiens & Lille)

Apr 05:

2^{nd} 3class groups G_{3}^{2}(K) of cyclic cubic fields K

of type (3,9)

and their connection with 2^{nd} 3Genus Groups g_{3}^{2}(K)

Mar 15:

Statistics of 2^{nd} 3class groups G_{3}^{2}(K) of cyclic cubic fields K

of type (3,3)

Frequency of Abelian, Coclass 1, and Coclass 2 Groups

Mar 08:

Statistics of 2^{nd} 2class groups G_{2}^{2}(K) of complex & real quadratic fields K

of type (2,2)

Frequency of Abelian, Dihedral, Quaternion, and Semidihedral Groups

Feb 25:

2^{nd} order IPADs τ_{p}^{2}(K) for fixed prime p = 2, resp. p = 3,

of complex quadratic fields

K = Q((1780)^{1/2}), K = Q((2067)^{1/2}), resp. K = Q((3896)^{1/2})

Feb 19:

Complete normal lattice of threestage 2tower groups G_{2}^{3}(K) of

type (2,4)

for complex quadratic fields K = Q((1780)^{1/2}) and K = Q((2067)^{1/2})

Feb 17:

The first 50 complex quadratic fields Q(D^{1/2}), 2408 ≤ D ≤ 260,

of type (2,4)

Transfer kernel and target types of 3 layers, result: 31 (62%) single stage 2towers

Feb 06:

Discovery of a possible 3class field tower group having three stages

for a TKT in Section F

namely of type F.11, first excited state, order 3^{29}, by M. F. Newman

Feb 03:

Discovery of possible 3class field tower groups having three stages

for TKTs in Section F

27 of each type F.7,11,12,13, ground state, order 3^{20}, by M. F. Newman

Jan 03:

Criteria for Principal Factorization Types of dihedral fields of degree 2p,

α_{1},α_{2},α_{3},β_{1},β_{2},γ,δ_{1},δ_{2},ε,

based on integral representations of Sunit groups by Alex Bartel (Coventry)

2012

Nov 23:

Transfer kernels and targets of metabelian pgroups of order p^{5}, type

(p,p), coclass 2, class 3

first commutator calculus for stem of isoclinism family Φ_{6}, p+7 isomorphism classes

Oct 30:

Discovery of further 3class field towers having three stages, such as

for k = Q((262744)^{1/2})

of type E.14 in its 1^{st} excited state, by M. R. Bush and D. C. Mayer

Oct 15:

Transfer kernels and targets of metabelian 7groups of order 16807

(7,7), coclass 2, class 3

first commutator calculus for stem of isoclinism family Φ_{6}, 14 isomorphism classes

Oct 11:

Nonmetabelian 3groups of order 3^{8}, class 5, coclass 3, derived length 3

which are Schur σgroups

analysis of their lower and upper central series and pcpresentations

Sep 17:

Jahrestagung der DMV, Saarbrücken, BRD

Presentation

Abstract

Aug 24:

3tower has length 3 for complex quadratic fields having same TKT/TTT

as k = Q((9748)^{1/2})

proved by N. Boston, M. R. Bush and D. C. Mayer

Aug 23:

First faultless disproof of Scholz/Taussky's claim of 3tower length 2

for k = Q((9748)^{1/2})

by M. R. Bush and D. C. Mayer

Aug 17:

G_{2}^{2}(k) of k = Q((D)^{1/2}) identified as descendant of <32,35>

for D = 9380, 14980

in a series of type (2,2,2) by Lemmermeyer, realizing family #79 by Newman/O'Brien

Aug 09:

Isomorphism of G_{3}^{2}(B) and G_{3}^{2}(k) for k = Q((D)^{1/2})

when k of type (3,3)

20% inherited by lifting the entire 3class field tower,
Strategic Operation on Nagasaki Day

Aug 06:

G_{3}^{2}(B) for B = Q(D^{1/2},(1)^{1/2}), GaussDirichlet type, D < 3*10^{4}

211 bicyclic biquadratic (3,3)

unique computation up to now,
Escalation of Nuclear War on Hiroshima Day

Jul 26:

Infinite prop Quantum Class Groups G_{p}^{∞}(K) of finite abelianization

Transfer Kernels & Targets

NonAbelian CohenLenstra asymptotics: Arrigoni/Bembom/Bartholdy/Boston/Bush/Hajir

Jul 11:

G_{2}^{2}(K) for 52 fields K = Q(D^{1/2},(1)^{1/2}) with 0 < D < 10^{4}

of type (2,2,2)

QCD (Quantum Chromo Dynamic) response to Zekhnini's thermonuclear bomb

Jun 14:

G_{5}^{2}(M) of 5dual cyclic quartic mirror images M = Q((ζζ^{1})D^{1/2})

of type (5,5)

of 34 real, resp. 4 complex quadratic fields Q(D^{1/2}),
16000 < D < 3500

Jun 07:

G_{3}^{2}(B) = <27,2> abelian for 79% of B = Q(D^{1/2},(3)^{1/2})

single stage 3tower for (9,3)

all 73 (100%), resp. 236/316 (75%),
where Cl_{3}Q(D^{1/2}) = (9), resp. (3)

Jun 06:

G_{3}^{2}(B) for B = Q(D^{1/2},(3)^{1/2}), D < 5*10^{4}

389 bicyclic biquadratic (9,3)

unique computation up to now (extension of Apr 07, 2012)

Jun 05:

G_{3}^{2}(B) of order 6561 for B = Q(D^{1/2},(3)^{1/2}), D = 20521, 40156

(3,3) TKT b.10 (0043)

unique two examples where predicted twisting of bipolarization actually occurs

May 31:

G_{3}^{2}(B) for B = Q(D^{1/2},(3)^{1/2}), D < 5*10^{4}

930 bicyclic biquadratic (3,3)

unique computation up to now (extension of Mar 07, 2012)

May 29:

Transfers of metabelian pgroups

Springer

Monatsh. Math. 166 (2012), no. 3  4, 467  495, DOI 10.1007/s006050100277x.

May 28:

Selection rules for triadic quantum class groups G = G_{3}^{2}(B)

of B = Q(D^{1/2},(3)^{1/2})

enforced by Kummer theory, Galois cohomology, automorphism groups Aut(G)

May 21:

G_{3}^{2}(B) = <729,34> for B = Q(2589^{1/2},(3)^{1/2})

(3,3) TKT b.10 (0043)

Ascione's nonCFgroup H, derived subgroup (3,3,3,3) instead of (9,3,3)

Apr 26:

International Workshop NTCCCS, Oujda, Morocco

Presentation

Apr 08:

G_{3}^{2}(B) = <729,37> for B = Q(2177^{1/2},(3)^{1/2})

(3,3) TKT b.10 (0043)

Ascione's nonCFgroup A, first triadic quantum class group of coclass 2

Apr 07:

G_{3}^{2}(B) for B = Q(D^{1/2},(3)^{1/2}), Eisenstein type, D < 3*10^{4}

213 bicyclic biquadratic (9,3)

unique computation up to now, πoλεμos πατηρ παντων

Mar 07:

G_{3}^{2}(B) for B = Q(D^{1/2},(3)^{1/2}), Eisenstein type, D < 3*10^{4}

549 bicyclic biquadratic (3,3)

unique computation up to now,
Initialization of Nuclear War "Ianna Bimetal"
(^{235}_{92}, ^{239}_{94})

Feb 24:

The second pclass group of a number field

World Scientific

Int. J. Number Theory 8 (2012), no. 2, 471  505, DOI 10.1142/S179304211250025X.

Jan 25:

G_{3}^{2}(K) = E↓^{4}, E↓^{2} for K = Q(D^{1/2}), D = 6540917, 8626101

(9,3) TKT b.15 (000;4)

second layer type (9,9,9,3), resp. (9,3,3,3), instead of (9,9,3,3)

2011

Dec 29:

G_{5}^{2}(K) for K = Q(D^{1/2}), D < 2*10^{7}

270 real (5,5)

unique computation up to now

Dec 23:

G_{5}^{2}(K) for K = Q(D^{1/2}), D > 2*10^{6}

813 complex (5,5)

unique computation up to now

Nov 04:

G_{5}^{2}(K) = <3125,14> for K = Q((89751)^{1/2})

(5,5) TKT (123456)

first realization of Taussky's fixed point principalization problem

Oct 31:

G_{3}^{2}(K) = E↓^{4} for K = Q(3763580^{1/2})

(9,3) TKT b.15 (000;4)

first realization of descendant of Ascione's CFgroup E = <243,13>

Oct 27:

G_{3}^{2}(K) = <2187,349> = H↓^{2} for K = Q(700313^{1/2})

(9,3) TKT b.15 (000;4)

first realization of descendant of Ascione's CFgroup H = <243,14>

Sep 25:

Joint CSASC Conference, Krems, Austria

Presentation

Jul 29:

G_{3}^{2}(K) for K = Q(D^{1/2}), D < 10^{7}

271 real (9,3)

unique computation up to now

Jul 23:

G_{3}^{2}(K) for K = Q(D^{1/2}), D > 10^{6}

875 complex (9,3)

by far most extensive computation up to now

Jul 22:

Transfer kernels and targets of metabelian 3groups

(9,3), coclass 2, class 3

first commutator calculus for branch 1 of isoclinism family Φ_{3}

Jul 09:

Transfer kernels and targets of metabelian 3groups

(9,3), coclass 3, class 3

first commutator calculus for branch 1 of isoclinism family Φ_{6}

Jul 01:

27^{th} Journées Arithmétiques, Vilnius, Lithuania

Presentation

Abstract

2010

Nov 08:

Transfer kernels and targets of metabelian 5groups of order 3125

(5,5), coclass 2, class 3

first commutator calculus for stem of isoclinism family Φ_{6}, 12 isomorphism classes

Sep 27:

Journées de Théorie des Nombres, Oujda, Morocco

Presentation

Reference

Jul 25:

G_{3}^{2}(K) for K = Q(D^{1/2}), D > 10^{6}

2020 complex (3,3)

by far most extensive computation up to now

Mar 31:

G_{3}^{2}(K) for K = Q(D^{1/2}), D < 10^{7}

2576 real (3,3)

by far most extensive computation up to now (extension of Dec 17, 2009)

Mar 20:

Transfer kernels of type d (.043) or d* (0.43) of metabelian 3groups

(3,3), coclass ≥ 3

supplementary commutator calculus to distinguish terminal and capable vertices

Mar 18:

G_{3}^{2}(K) for K = Q(8491713^{1/2})

(3,3) TKT d*.25 (0143)

first triadic quantum class group of coclass 4 over real quadratic base field

2009

Dec 17:

G_{3}^{2}(K) for K = Q(D^{1/2}), D < 10^{6}

149 real (3,3)

by far most extensive computation up to now

Dec 07:

Top Down Principalization Algorithm

via class group structure

complete determination of 3class groups of type (3,3,3) of unramified S_{3} fields

Nov 09:

G_{3}^{2}(K) for K = Q(710652^{1/2})

(3,3) TKT b.10 (0043)

first triadic quantum class group of coclass 3 over real quadratic base field

Nov 04:

Transfer kernels of metabelian pgroups

(p,p), coclass 1

first commutator calculus for primes p greater than or equal to 5

Nov 02:

Transfer kernels of metabelian 2groups

(2,2), coclass 1

first explicit commutator calculus for exceptional prime p = 2

Sep 25:

Jahrestagung der ÖMG und DMV, Graz, Austria

Presentation

Abstract

2008

Jan 01:

G_{3}^{2}(K) = <729,54> for K = Q(540365^{1/2})

(3,3) TKT c.21 (0231)

first realization of capable vertex by triadic quantum class group over real quadratic field

2006

Jan 30:

G_{3}^{2}(K) = <729,57> for K = Q(214712^{1/2})

(3,3) TKT G.19 (2143)

first triadic quantum class group of coclass 2 over real quadratic base field
