Research Frontier 2013

Reaching Current Limits & Breaking Through Beyond



Karl-Franzens University Graz, left side

Reaching Current Limits:

Karl-Franzens University Graz, centre with 8 figures

Breaking Through Beyond:

  • *****************************************************
    Since Thursday, August 23, 2012, 01:06 MET,
    BUSH and MAYER were the first who proved that
    the COMPLEX QUADRATIC FIELD K = Q((-9748)1/2)
    has 3-CLASS FIELD TOWER of LENGTH AT LEAST THREE.
    *****************************************************

  • Proof:
    (1) Daniel C. MAYER proved that
    the metabelianization G/G'' of the 3-tower group G of K = Q((-9748)1/2),
    and in fact of any algebraic number field
    with second 3-class group g having
    transfer kernel type (TKT) E.9, κ(g) = (2,2,3,1), and
    transfer target type (TTT) τ(g) = [(9,27),(3,9)3],
    is one of two terminal metabelian vertices SmallGroup (2187,302) or SmallGroup(2187,306)
    on branch B( SmallGroup(729,54) ) of coclass tree T( SmallGroup(243,8) ).
    (2) Michael R. BUSH computed a list of all Schur σ-groups of order 37 = 2187
    which only contains SmallGroup(2187,121) and SmallGroup(2187,122)
    but neither SmallGroup(2187,302) nor SmallGroup(2187,306).
    (3) Both authors exchanged and combined their results at 01:06 MET on August 23, 2012.
    Since K is complex quadratic, its 3-tower group G must be a Schur σ-group,
    according to Shafarevich , Koch and Venkov .
    Since none of the groups SmallGroup(2187,302) and SmallGroup(2187,306) is a Schur σ-group,
    the 3-tower of K cannot stop at the second stage and
    G must be a non-metabelian group of derived length at least 3.
    QED.

  • We met at the
    Workshop on "GOLOD SHAFAREVICH theory" ,
    organized by the Erwin Schrödinger Institute (ESI),
    Boltzmanngasse 9, 1090 VIENNA.

    After a brief discussion of about an hour,
    we found the solutions of problems
    that mathematicians were unable to answer for nearly 80 years
    ,
    since Scholz and Taussky claimed that the mentioned fields
    have 3-towers of length 2 in their famous paper of 1934.

  • In this manner, we will continue
    to shed light on the completely unsolved question of
    3-stage and higher towers of p-class fields for odd primes p
    by determining exact borders between vertices of
    different derived length on coclass graphs G(p,r),
    and investigating the second derived quotient G/G''
    of vertices G with derived length dl(G) ≥ 3.





Karl-Franzens University Graz, right side

Please Note:

  • This is the BEGINNING of a new era of research concerning
    the maximal unramified pro-p extensions of number fields
    by joining coclass theory of finite p-groups and
    suitable generalizations of Schur σ-groups.

    And it's the END of intolerable uncertainty since 1933.

Daniel C. Mayer
Our international research project
Towers of p-Class Fields
over Algebraic Number Fields

is supported by the
Public access to our papers and presentations
is granted via
Our services to the mathematical community
are listed in
Presentations and Lectures 2013:


Bibliographical References:

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Capitulation des 2-classes d'idéaux de Q(21/2,d1/2) où d est un entier naturel sans facteurs carrés,
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Capitulation des 2-classes d'idéaux de k=Q((2p)1/2,i),
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Capitulation dans le corps des genres de certain corps de nombres biquadratique imaginaire
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,
preprint, 2010.

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Maximal unramified 3-extensions of imaginary quadratic fields and SL2Z3,
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Topics in computational algebraic number theory,
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The capitulation problem in class field theory,
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SmallGroups - a library of groups of small order,
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The Magma algebra system. I. The user language,
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[BCFS] W. Bosma, J. J. Cannon, C. Fieker, and A. Steels (eds.),
Handbook of Magma functions,
Edition 2.19, Sydney, 2013.

[BBH] N. Boston, M. R. Bush and F. Hajir,
Heuristics for p-class towers of imaginary quadratic fields,
arXiv:1111.4679v1 [math.NT] (2011).

[Br] J. R. Brink,
The class field tower for imaginary quadratic number fields of type (3,3),
Dissertation, Ohio State Univ., 1984.

[BrGo] J. R. Brink and R. Gold,
Class field towers of imaginary quadratic fields,
manuscripta math. 57 (1987), 425 - 450.

[Bu] M. R. Bush,
Computation of Galois groups associated to the 2-class towers of some quadratic fields,
J. Number Theory 100 (2003), 313 - 325.

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Capitulation in class field extensions of type (p,p),
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Un problème de capitulation,
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[Dh] A. Derhem,
Capitulation dans les extensions quadratiques non ramifiées de corps de nombres cubiques cycliques,
Thèse de doctorat, Université Laval, Québec, 1988.

[DEF] H. Dietrich, B. Eick, and D. Feichtenschlager,
Investigating p-groups by coclass with GAP,
Computational group theory and the theory of groups, pp. 45 - 61, Contemp. Math. 470, AMS, Providence, RI, 2008.

[Dt1] H. Dietrich,
Periodic patterns in the graph of p-groups of maximal class,
J. Group Theory 13 (2010), 851 - 871.

[Dt2] H. Dietrich,
A new pattern in the graph of p-groups of maximal class,
Bull. London Math. Soc. 42 (2010), 1073 - 1088.

[dS] M. du Sautoy,
Counting p-groups and nilpotent groups,
Inst. Hautes Études Sci. Publ. Math. 92 (2001), 63 - 112.

[Ef] T. E. Easterfield,
A classification of groups of order p6,
Ph. D. Thesis, Univ. of Cambridge, 1940.

[EkLg] B. Eick and C. Leedham-Green,
On the classification of prime-power groups by coclass,
Bull. London Math. Soc. 40 (2) (2008), 274 - 288.

[EkFs] B. Eick and D. Feichtenschlager,
Infinite sequences of p-groups with fixed coclass
(preprint 2010).

[ELNO] B. Eick, C. R. Leedham-Green, M. F. Newman, and E. A. O'Brien,
On the classification of groups of prime-power order by coclass: The 3-groups of coclass 2
(preprint 2011).

[EnTu1] V. Ennola and R. Turunen,
On totally real cubic fields,
Math. Comp. 44 (1985), 495 - 518.

[EnTu2] V. Ennola and R. Turunen,
On cyclic cubic fields,
Math. Comp. 45 (1985), 585 - 589.

[Fi] C. Fieker,
Computing class fields via the Artin map,
Math. Comp. 70 (2001), no. 235, 1293 - 1303.

[Fu1] Ph. Furtwängler,
Über das Verhalten der Ideale des Grundkörpers im Klassenkörper,
Monatsh. Math. Phys. 27 (1916), 1 - 15.

[Fu2] Ph. Furtwängler,
Beweis des Hauptidealsatzes für die Klassenkörper algebraischer Zahlkörper,
Abh. Math. Sem. Univ. Hamburg 7 (1929), 14 - 36.

[Fu3] Ph. Furtwängler,
Über eine Verschärfung des Hauptidealsatzes für algebraische Zahlkörper,
J. Reine Angew. Math. 167 (1932), 379 - 387.

[GNO] G. Gamble, W. Nickel, and E. A. O'Brien,
ANU p-Quotient --- p-Quotient and p-Group Generation Algorithms,
2006, an accepted GAP 4 package, available also in MAGMA.

[GAP] The GAP Group,
GAP - Groups, Algorithms, and Programming -
a System for Computational Discrete Algebra, Version 4.4.12
,
Aachen, Braunschweig, Fort Collins, St. Andrews, 2008, http://www.gap-system.org

[Ge] F. Gerth III,
Ranks of 3-class groups of non-Galois cubic fields,
Acta Arith. 30 (1976), 307 - 322.

[Gr] G. Gras,
Sur les l-classes d'idéaux dans les extensions cycliques relatives de degré premier l,
Ann. Inst. Fourier, Grenoble 23 (1973), no. 4, 1 - 44.

[Gr1] G. Gras,
Sur les l-classes d'idéaux des extensions non galoisiennes de degré premier impair l
à la clôture galoisienne diédrale de degré 2l
,
J. Math. Soc. Japan 26 (1974), 677 - 685.

[Gr2] M.-N. Gras,
Méthodes et algorithmes pour le calcul numérique du nombre de classes et des unités
des extensions cubiques cycliques de Q
,
J. reine angew. Math. 277 (1975), 89 - 116.

[Hl] Ph. Hall,
The classification of prime-power groups,
J. reine angew. Math. 182 (1940), 130 - 141.

[Ha3] H. Hasse,
Arithmetische Theorie der kubischen Zahlkörper auf klassenkörpertheoretischer Grundlage,
Math. Z. 31 (1930), 565 - 582.

[HeSm] F.-P. Heider und B. Schmithals,
Zur Kapitulation der Idealklassen in unverzweigten primzyklischen Erweiterungen,
J. reine angew. Math. 336 (1982), 1 - 25.

[Hi1] D. Hilbert,
Über den Dirichlet'schen biquadratischen Zahlkörper,
Math. Annalen 45 (1894), 309 - 340.

[Hi2] D. Hilbert,
Die Theorie der algebraischen Zahlkörper,
Jber. der D. M.-V. 4 (1897), 175 - 546.

[Ho] T. Honda,
Pure cubic fields whose class numbers are multiples of three,
J. Number Theory 3 (1971), 7 - 12.

[Is] M. C. Ismaïli,
Sur la capitulation des 3-classes d'idéaux de la clôture normale d'un corps cubique pur,
Thèse de doctorat, Université Laval, Québec, 1992.

[IsMe1] M. C. Ismaïli et R. El Mesaoudi,
Sur la divisibilité exacte par 3 du nombre de classes de certain corps cubiques purs,
Ann. Sci. Math. Québec 25 (2001), no. 2, 153 - 177.

[IsMe2] M. C. Ismaïli et R. El Mesaoudi,
Sur la capitulation des 3-classes d'idéaux de la clôture normale de certaines corps cubiques purs,
Ann. Sci. Math. Québec 29 (2005), no. 1, 49 - 72.

[Jm] R. James,
The groups of order p6 (p an odd prime),
Math. Comp. 34 (1980), no. 150, 613 - 637.

[Ki1] H. H. Kisilevsky,
Some results related to Hilbert's theorem 94,
J. number theory 2 (1970), 199 - 206.

[Ki2] H. H. Kisilevsky,
Number fields with class number congruent to 4 mod 8 and Hilbert's theorem 94,
J. number theory 8 (1976), 271 - 279.

[KoVe] H. Koch und B. B. Venkov,
Über den p-Klassenkörperturm eines imaginär-quadratischen Zahlkörpers,
Astérisque, 24 - 25 (1975), 57 - 67.

[Ku1] T. Kubota,
Über die Beziehung der Klassenzahlen der Unterkörper des bizyklischen biquadratischen Zahlkörpers,
Nagoya Math. J. 6 (1953), 119 - 127.

[Ku2] T. Kubota,
Über den bizyklischen biquadratischen Zahlkörper,
Nagoya Math. J. 10 (1956), 65 - 85.

[Kd] S. Kuroda,
Über den Dirichletschen Körper,
J. Fac. Sci. Imp. Univ. Tokyo, Sec. I, Vol. 4 (1943), Part 5, 383 - 406.

[LhCh] Ou. Lahlou et M. Charkani El Hassani,
Arithmétique d'une famille de corps cubiques,
C. R. Math. Acad. Sci. Paris 336 (2003), no. 5, 371 - 376.

[LgMk] C. R. Leedham-Green and S. McKay,
The structure of groups of prime power order,
London Math. Soc. Monographs, New Series, 27, Oxford Univ. Press, 2002.

[LgNm] C. R. Leedham-Green and M. F. Newman,
Space groups and groups of prime power order I,
Arch. Math. 35 (1980), 193 - 203.

[Lm] F. Lemmermeyer,
Class groups of dihedral extensions,
Math. Nachr. 278 (2005), no. 6, 679 - 691.

[Lr] A. Leriche,
Cubic, quartic and sextic Pólya fields,
J. Number Theory 133 (2013), 59 - 71.

[MAGMA] The MAGMA Group,
MAGMA Computational Algebra System, Version 2.19-5,
Sydney, 2013, http://magma.maths.usyd.edu.au

[Mt] J. Martinet,
Sur l'arithmétique des extensions galoisiennes à groupe de galois diédral d'ordre 2p,
Ann. Inst Fourier, Grenoble 19 (1963), 1 - 80.

[MtPn] J. Martinet et J.-J. Payan,
Sur les extensions cubiques non-Galoisiennes de rationels et leur clôture Galoisienne,
J. reine angew. Math. 228 (1965), 15 - 37.

[Ma] D. C. Mayer,
Multiplicities of dihedral discriminants,
Math. Comp. 58 (1992), no. 198, 831-847, supplements section S55-S58, DOI 10.2307/2153221.

[Ma0] D. C. Mayer,
Discriminants of metacyclic fields,
Canad. Math. Bull. 36(1) (1993), 103-107.

[Ma1] D. C. Mayer,
Principalization in complex S3-fields,
Congressus Numerantium 80 (1991), 73 - 87,
Proceedings of the Twentieth Manitoba Conference on Numerical Mathematics and Computing, Winnipeg, Manitoba, 1990.

[Ma2] D. C. Mayer,
Transfers of metabelian p-groups,
Monatsh. Math. 166 (2012), no. 3 - 4, 467 - 495, DOI 10.1007/s00605-010-0277-x.

[Ma3] 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.

[Ma4] D. C. Mayer,
Principalization algorithm via class group structure
(preprint 2011).

[Ma5] D. C. Mayer,
The distribution of second p-class groups on coclass graphs,
J. Théor. Nombres Bordeaux 25 (2013), no. 2, 401 - 456.
27th Journées Arithmétiques, Faculty of Mathematics and Informatics, Vilnius University, Vilnius, Lithuania, 2011.

[Ma6] D. C. Mayer,
Lattice minima and units in real quadratic number fields,
Publicationes Mathematicae Debrecen
39 (1991), 19-86.

[Ma7] D. C. Mayer,
Differential principal factors and units in pure cubic number fields,
Dept. of Math., Univ. Graz, 1989.

[Ma8] D. C. Mayer,
Classification of dihedral fields,
Dept. of Computer Science, Univ. of Manitoba, 1991.

[Ma9] D. C. Mayer,
List of discriminants dL<200000 of totally real cubic fields L,
arranged according to their multiplicities m and conductors f
,
Dept. of Computer Science, Univ. of Manitoba, 1991.

[Ma10] D. C. Mayer,
Principal Factorization Types of Multiplets of Pure Cubic Fields Q( R1/3 ) with R < 106,
Univ. Graz, Computer Centre, 2002.

[Ma11] D. C. Mayer,
Class Numbers and Principal Factorizations of Families of Cyclic Cubic Fields with Discriminant d < 1010,
Univ. Graz, Computer Centre, 2002.

[Ma12] D. C. Mayer,
Principalization in Unramified Cyclic Cubic Extensions
of all Quadratic Fields with Discriminant -50000 < d < 0 and 3-Class Group of Type (3,3)
,
Univ. Graz, Computer Centre, 2003

[Ma13] D. C. Mayer,
Principalization in Unramified Cyclic Cubic Extensions
of selected Quadratic Fields with Discriminant -200000 < d < -50000 and 3-Class Group of Type (3,3)
,
Univ. Graz, Computer Centre, 2004

[Ma14] D. C. Mayer,
Two-Stage Towers of 3-Class Fields over Quadratic Fields,
Univ. Graz, 2006.

[Ma15] D. C. Mayer,
3-Capitulation over Quadratic Fields with Discriminant |d| < 3*105 and 3-Class Group of Type (3,3),
Univ. Graz, Computer Centre, 2006.

[Ma16] D. C. Mayer,
Quadratic p-ring spaces for counting dihedral fields,
Dept. of Computer Science, Univ. of Manitoba, 2009.

[mL] C. McLeman,
p-tower groups over quadratic imaginary number fields,
Ann. Sci. Math. Québec 32 (2008), no. 2, 199 - 209.

[Mi] R. J. Miech,
Metabelian p-groups of maximal class,
Trans. Amer. Math. Soc. 152 (1970), 331 - 373.

[My] K. Miyake,
Algebraic investigations of Hilbert's Theorem 94, the principal ideal theorem and the capitulation problem,
Expo. Math. 7 (1989), 289 - 346.

[Mo] N. Moser,
Unités et nombre de classes d'une extension Galoisienne diédrale de Q,
Abh. Math. Sem. Univ. Hamburg 48 (1979), 54 - 75.

[Ne1] B. Nebelung,
Klassifikation metabelscher 3-Gruppen mit Faktorkommutatorgruppe vom Typ (3,3)
und Anwendung auf das Kapitulationsproblem
,
Inauguraldissertation, Band 1, Univ. zu Köln, 1989.

[Ne2] B. Nebelung,
Anhang zu Klassifikation metabelscher 3-Gruppen mit Faktorkommutatorgruppe vom Typ (3,3)
und Anwendung auf das Kapitulationsproblem
,
Inauguraldissertation, Band 2, Univ. zu Köln, 1989.

[Nm] M. F. Newman,
Groups of prime-power order
Groups - Canberra 1989, Lecture Notes in Mathematics, vol. 1456, 1990, pp. 49 - 62.

[NmOb] M. F. Newman and E. A. O'Brien,
Classifying 2-groups by coclass,
Trans. Amer. Math. Soc. 351 (1999), 131 - 169.

[Ob] E. A. O'Brien,
The p-group generation algorithm,
J. Symbolic Comput. 9 (1990), 677 - 698.

[PARI] The PARI Group,
PARI/GP, Version 2.3.4
Bordeaux, 2008, http://pari.math.u-bordeaux.fr

[Pa] Ch. J. Parry,
Bicyclic Bicubic Fields,
Canad. J. Math. 42 (1990), no. 3, 491 - 507.

[Re] H. Reichardt,
Arithmetische Theorie der kubischen Zahlkörper als Radikalkörper,
Monatsh. Math. Phys. 40 (1933), 323 - 350.

[Sm] B. Schmithals,
Kapitulation der Idealklassen und Einheitenstruktur in Zahlkörpern,
J. Reine Angew. Math. 358 (1985), 43 - 60.

[So1] A. Scholz,
Über die Beziehung der Klassenzahlen quadratischer Körper zueinander,
J. reine angew. Math. 166 (1932), 201 - 203.

[So2] A. Scholz,
Idealklassen und Einheiten in kubischen Körpern,
Monatsh. Math. Phys. 40 (1933), 211 - 222.

[SoTa] A. Scholz und O. Taussky,
Die Hauptideale der kubischen Klassenkörper imaginär quadratischer Zahlkörper:
ihre rechnerische Bestimmung und ihr Einfluß auf den Klassenkörperturm,
J. reine angew. Math. 171 (1934), 19 - 41.

[Sr1] O. Schreier,
Über die Erweiterung von Gruppen. I,
Monatsh. Math. Phys. 34 (1926), 165 - 180.

[Sr2] O. Schreier,
Über die Erweiterung von Gruppen. II,
Hamburg. Sem. Abh. 4 (1926), 321 - 346.

[Sh] I. R. Shafarevich,
Rasshireniya s zadannymi tochkami vetvleniya,
(Extensions with prescribed ramification points),
Inst. Hautes Études Sci. Publ. Math. 18 (1963), 71 - 95 (Russian).

[SnKw] J.-J. Son and S.-H. Kwon,
On the principal ideal theorem,
J. Korean Math. Soc. 44 (2007), no. 4, 747 - 756.

[Su] H. Suzuki,
A generalization of Hilbert's Theorem 94,
Nagoya Math. J. 121 (1991), 161 - 169.

[Tl] M. Talbi,
Capitulation des 3-classes d'idéaux dans certains corps de nombres,
Thèse de doctorat, Université Mohammed Premier, Oujda, Morocco, 2008.

[Ta] O. Taussky,
Über eine Verschärfung des Hauptidealsatzes für algebraische Zahlkörper,
J. Reine Angew. Math. 168 (1932), 193 - 210.

[Ta1] O. Taussky,
A remark on the class field tower,
J. London Math. Soc. 12 (1937), 82 - 85.

[Ta2] O. Taussky,
A remark concerning Hilbert's Theorem 94,
J. Reine Angew. Math. 239/240 (1970), 435 - 438.

[Vo1] G. F. Voronoi,
O celykh algebraicheskikh chislakh zavisyashchikh ot kornya uravneniya tretei stepeni
(On the algebraic integers derived from a root of a third degree equation),
Master's thesis, 1894, St. Peterburg (Russian).

[Vo2] G. F. Voronoi,
Ob odnom obobshchenii algorifma nepreryvnykh drobei
(On a generalization of the algorithm of continued fractions),
Doctoral Dissertation, 1896, Warsaw (Russian).

[Yo] E. Yoshida,
On the 3-class field tower of some biquadratic fields,
Acta Arith. 107 (2003), no. 4, 327 - 336.

[Wa1] H. Wada,
On cubic Galois extensions of Q( (-3)1/2 ),
Proc. Japan Acad. 46 (1970), 397 - 400.

[Wa2] H. Wada,
A table of ideal class groups of imaginary quadratic fields,
Proc. Japan Acad. 46 (1970), 401 - 403.

[Wa3] H. Wada,
A table of fundamental units of purely cubic fields,
Proc. Japan Acad. 46 (1970), 1135 - 1140.

[Wi3] H. C. Williams,
Determination of principal factors in Q(D1/2) and Q(D1/3),
Math. Comp. 38 (1982), no. 157, 261 - 274.

*
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*

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