APM  Vol.5 No.5 , April 2015
Index-p Abelianization Data of p-Class Tower Groups
Abstract: Given a fixed prime number p, the multiplet of abelian type invariants of the p-class groups of all unramified cyclic degree p extensions of a number field K is called its IPAD (index-p abeliani- zation data). These invariants have proved to be a valuable information for determining the Galois group of the second Hilbert p-class field and the p-capitulation type of K. For p=3 and a number field K with elementary p-class group of rank two, all possible IPADs are given in the complete form of several infinite sequences. Iterated IPADs of second order are used to identify the group of the maximal unramified pro-p extension of K.
Cite this paper: Mayer, D. (2015) Index-p Abelianization Data of p-Class Tower Groups. Advances in Pure Mathematics, 5, 286-313. doi: 10.4236/apm.2015.55029.

[1]   The PARI Group (2014) PARI/GP. Version 2.7.2, Bordeaux.

[2]   Bosma, W., Cannon, J. and Playoust, C. (1997) The Magma Algebra System I: The User Language. Journal of Symbolic Computation, 24, 235-265.

[3]   Bosma, W., Cannon, J.J., Fieker, C. and Steels, A., Eds. (2015) Handbook of Magma Functions. Edition 2.21, University of Sydney, Sydney.

[4]   The MAGMA Group (2015) MAGMA Computational Algebra System. Version 2.21-2, Sydney.

[5]   Mayer, D.C. (2014) Principalization Algorithm via Class Group Structure. Journal de Théorie des Nombres de Bordeaux, 26, 415-464.

[6]   Bush, M.R. and Mayer, D.C. (2015) 3-Class Field Towers of Exact Length 3. Journal of Number Theory, 147, 766-777.

[7]   Artin, E. (1927) Beweis des allgemeinen Reziprozitatsgesetzes. Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg, 5, 353-363.

[8]   Artin, E. (1929) Idealklassen in Oberkorpern und allgemeines Reziprozitatsgesetz. Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg, 7, 46-51.

[9]   Mayer, D.C. (2013) The Distribution of Second p-Class Groups on Coclass Graphs. Journal de Théorie des Nombres de Bordeaux, 25, 401-456.

[10]   Boston, N., Bush, M.R. and Hajir, F. (2014) Heuristics for p-Class Towers of Imaginary Quadratic Fields, with an Appendix by Jonathan Blackhurst. Mathematische Annalen, in Press.

[11]   Boston, N. and Leedham-Green, C. (2002) Explicit Computation of Galois p-Groups Unramified at p. Journal of Algebra, 256, 402-413.

[12]   Bush, M.R. (2003) Computation of Galois Groups Associated to the 2-Class Towers of Some Quadratic Fields. Journal of Number Theory, 100, 313-325.

[13]   Bartholdi, L. and Bush, M.R. (2007) Maximal Unramified 3-Extensions of Imaginary Quadratic Fields and SL2(Z3). Journal of Number Theory, 124, 159-166.

[14]   Boston, N. and Nover, H. (2006) Computing Pro-p Galois Groups. Proceedings of ANTS 2006, Lecture Notes in Computer Science 4076, Springer-Verlag Berlin Heidelberg, Berlin, 1-10.

[15]   Nover, H. (2009) Computation of Galois Groups of 2-Class Towers. Ph.D. Thesis, University of Wisconsin, Madison.

[16]   Mayer, D.C. (2015) Periodic Bifurcations in Descendant Trees of Finite p-Groups. Advances in Pure Mathematics, 5, 162-195.

[17]   Besche, H.U., Eick, B. and O’Brien, E.A. (2002) A Millennium Project: Constructing Small Groups. International Journal of Algebra and Computation, 12, 623-644.

[18]   Besche, H.U., Eick, B. and O’Brien, E.A. (2005) The Small Groups Library—A Library of Groups of Small Order. An Accepted and Refereed GAP 4 Package, Available Also in MAGMA.

[19]   Mayer, D.C. (2012) Transfers of Metabelian p-Groups. Monatshefte für Mathematik, 166, 467-495.

[20]   Scholz, A. and Taussky, O. (1934) Die Hauptideale der kubischen Klassenkorper imaginar quadratischer Zahlkorper: Ihre rechnerische Bestimmung und ihr Einflub auf den Klassenkorperturm. Journal für die Reine und Angewandte Mathematik, 171, 19-41.

[21]   Mayer, D.C. (1990) Principalization in Complex S3-Fields. Congressus Numerantium, 80, 73-87.

[22]   Taussky, O. (1970) A Remark Concerning Hilbert’s Theorem 94. Journal für die Reine und Angewandte Mathematik, 239/240, 435-438.

[23]   Mayer, D.C. (2012) The Second p-Class Group of a Number Field. International Journal of Number Theory, 8, 471-505.

[24]   Gamble, G., Nickel, W. and O’Brien, E.A. (2006) ANU p-Quotient—p-Quotient and p-Group Generation Algorithms. An Accepted GAP 4 Package, Available Also in MAGMA.

[25]   The GAP Group (2015) GAP—Groups, Algorithms, and Programming—A System for Computational Discrete Algebra. Version 4.7.7, Aachen, Braunschweig, Fort Collins, St. Andrews.

[26]   Ascione, J.A., Havas, G. and Leedham-Green, C.R. (1977) A Computer Aided Classification of Certain Groups of Prime Power Order. Bulletin of the Australian Mathematical Society, 17, 257-274, Corrigendum 317-319, Microfiche Supplement, 320.

[27]   Ascione, J.A. (1979) On 3-Groups of Second Maximal Class. Ph.D. Thesis, Australian National University, Canberra.

[28]   O’Brien, E.A. (1990) The p-Group Generation Algorithm. Journal of Symbolic Computation, 9, 677-698.

[29]   Holt, D.F., Eick, B. and O’Brien, E.A. (2005) Handbook of Computational Group Theory. Discrete Mathematics and Its Applications, Chapman and Hall/CRC Press.

[30]   Heider, F.-P. and Schmithals, B. (1982) Zur Kapitulation der Idealklassen in unverzweigten primzyklischen Erweiterungen. Journal für die reine und angewandte Mathematik, 336, 1-25.

[31]   Shafarevich, I.R. (1966) Extensions with Prescribed Ramification Points. Publications mathématiques de l’IHéS, 18, 71-95.

[32]   Koch, H. and Venkov, B.B. (1975) über den p-Klassenkorperturm eines imaginar-quadratischen Zahlkorpers. Astérisque, 24-25, 57-67.

[33]   Diaz, F. and Diaz (1973/74) Sur les corps quadratiques imaginaires dont le 3-rang du groupe des classes est supérieur à 1. Séminaire Delange-Pisot-Poitou, No. G15.

[34]   Diaz, F. and Diaz (1978) Sur le 3-rang des corps quadratiques. No. 78-11, Université Paris-Sud, Orsay.

[35]   Buell, D.A. (1976) Class Groups of Quadratic Fields. Mathematics of Computation, 30, 610-623.

[36]   Golod, E.S. and Shafarevich, I.R. (1965) On Class Field Towers. Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya, 28, 261-272.

[37]   Vinberg, E.B. (1965) On a Theorem Concerning the Infinite-Dimensionality of an Associative Algebra. Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya, 29, 209-214.

[38]   Shanks, D. (1976) Class Groups of the Quadratic Fields Found by Diaz y Diaz. Mathematics of Computation, 30, 173-178.

[39]   Sloane, N.J.A. (2014) The On-Line Encyclopedia of Integer Sequences (OEIS). The OEIS Foundation Inc.

[40]   Fieker, C. (2001) Computing Class Fields via the Artin Map. Mathematics of Computation, 70, 1293-1303.

[41]   Mayer, D.C. (2015) Index-p Abelianization Data of p-Class Tower Groups. Proceedings of the 29th Journées Arithmé-tiques, Debrecen, 6-10 July 2015, in Preparation.