P26008-N25 Towers of p-class fields over algebraic number fields
 
Project
Abstract
Scientists/Scholars
Project Publications
Further Activities
Cooperations
Final Report

Further Activities

Description and Links to further activities

(1)
Web Site www.algebra.at
Since June 1999, I am owner of the internet domain www.algebra.at, registered by Internet Network Information Center (InterNIC), and I am operating an own web server to present my research results in algebraic number theory and p-group theory, within the scope of annual banners devoted to various special topics, and to grant free public access to my scientific databases which provide considerably more information than can be published in printed form.

My databases have proved to be useful for other researchers, e. g., my detailed tables on transfer kernel types were used in the Ph. D. Thesis of Tobias Bembom written under supervision of Preda Mihailescu, Georg-August University, Göttingen, 2012, pp. 5 and 126.

A particular survey on cubic field extensions was created in 2003 for the Number Theory Web, following an invitation by Keith Matthews at Brisbane.

The annual banner for 2014 with title Fame for Styria 2014 is devoted to the stand-alone research project P26008-N25 in honour of my home town Graz, the capital of Styria. As a flagship topic, I have selected the following presentation on pure absolute extensions of degree five.

(1.1)
Pure quintic number fields:
The second research line of the project P26008-N25 is concerned with ramified cyclic relative extensions N/K of odd prime degree p ≥ 3 with non-trivial conductor c = f(N/K) > 1, located within the p-ring class field modulo c, Fp,c(K), of a quadratic or cyclotomic field K. With my papers Multiplicities of dihedral discriminants and Classification of dihedral fields and Discriminants of metacyclic fields, written in the course of my Schrödinger project in Canada, I provided an extensible foundation for all aspects of ramification, which were enlightened deeper in my presentation Multiplicities of discriminants of p-ring class fields over quadratic number fields with modified p-class rank σ ≥ 2 and were brought to a preliminary completion for σ = 2 in two presentations, Quadratic p-ring spaces for counting dihedral fields and Number fields sharing a common discriminant. In the present project I am going to extend these results to pure quintic fields.

Chapter I
§ 1. Galois closure and subfield lattice
§ 2. Class numbers and unit index

Chapter II
§ 3. Conductor and discriminants
§ 4. Herbrand quotient of units

Chapter III
§ 5. Ambiguous principal ideals
§ 6. Differential principal factorizations

Chapter IV
§ 7. Types of pure quintic fields
§ 8. Computational results

Chapter V
§ 9. Algorithmic techniques
§ 10. Theoretical foundations

(1.2)
Pure cubic number fields:
To enable a detailed comparison with the less complex but nevertheless very interesting case of pure cubic fields, I have developed a parallel presentation which focusses these objects from the most up-to-date point of view:

Chapter I
§ 1. Galois closure and subfield lattice
§ 2. Class numbers and unit index
§ 3. Conductor and discriminants

Chapter II
§ 4. Ambiguous principal ideals
§ 5. Differential principal factorizations

Chapter III
§ 6. Types of pure cubic fields
§ 7. Computational results
§ 8. Theoretical results

(2)
Contributions to Wikipedia
(2.1)
Descendant tree (group theory).
(2.2)
Artin transfer (group theory).
(2.3)
Principalization (algebra).
(2.4)
P-group generation algorithm.
(2.5)
Induced homomorphism (quotient group).

(3)
Contributions to the Online Encyclopedia of Integer Sequences (OEIS) by Neil Sloane
(3.1)
Positive quadratic discriminants of fields with 3-class rank 1:
3-class groups of type (3);
(3.2)
Discriminants of biquadratic fields with 3-class rank 2:
3-class groups of type (3,3);
Position of second 3-class groups on coclass graph G(3,1):
Second 3-class group <9,2>;
Second 3-class group <81,9>;
Second 3-class group <729,95>;
Position of second 3-class groups on coclass graph G(3,2):
Second 3-class group <729,37>;
Second 3-class group <729,34>;
(3.3)
Positive quadratic discriminants of fields with 3-class rank 2:
3-class groups of rank 2;
3-class groups of type (3,3);
3-class field towers of length 2:
Principalization type a.2;
Principalization type a.3;
3-class field towers of length 3:
Principalization type c.18;
Principalization type c.21;
(3.4)
Negative quadratic discriminants of fields with 3-class rank 3,
giving rise to 3-class field towers of infinite length:
3-class groups of rank 3,
3-class groups of type (3,3,3).
(3.5)
Negative quadratic discriminants of fields with 3-class rank 2
3-class groups of rank 2,
3-class groups of type (3,3);
Position of second 3-class groups on coclass graph G(3,2):
Sporadic second 3-class groups outside of coclass trees,
Periodic second 3-class groups on coclass trees;
3-class field towers of length 2:
Principalization types D.5, D.10,
Principalization type D.10,
Principalization type D.5;
3-class field towers of length 3:
Principalization types E.6, E.14, E.8, E.9,
Excited states of type E.6,
Excited states of type E.14,
Excited states of type E.8,
Excited states of type E.9;
3-class field towers of unknown length:
Principalization type H.4*,
Principalization type G.19,
Excited states of type H.4,
Excited states of type G.16.

(3.6)
Clusters of squarefree integers with fixed number of prime divisors:
(3.6.1)
Triplets:
increasing number of primes.
(3.6.2)
Sextets with central gap:
increasing number of primes;
two primes,
three primes,
four primes,
five primes.

(3.7)
Sequences associated with Eric S. Rowland's prime-generating sequence:
Sequence of first differences,
Subsequence of actual primes.

(4)
Contributions to the Great Internet Mersenne Prime Search (GIMPS) by George Woltman
(Mersenne numbers 2p-1 with prime exponents p)
After only 13 months of activity,
I conquered ranks among the top 2% of all participants:
(4.1)
Lucas Lehmer tests (using FFT and DFT):
top producers rank 32 among 3014.
(4.2)
Total overall:
top producers rank 124 among 6717.
  
 
 

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