Scientific Research 2010



Navigation Map through our Current Research Lines

their Historical Origins, and Future Developments©


The aim of the following tree diagram with links to the subsequent bibliographical references is to provide an overview of our current research lines,
their historical origins, and future developments©.
For each research line we give the 2010 Mathematics Subject Classification (MSC) of the American Mathematical Society (AMS).
  • Line 5: Class numbers, class groups, and discriminants (11R29), Class field theory (11R37), Nilpotent groups and p-groups (20D15),
    Commutator calculus (20F12), Derived and central series (20F14).
  • Line 4: Algebraic number theory: global fields (11Rxx), Quadratic extensions (11R11), Cubic and quartic extensions (11R16),
    Other abelian and metabelian extensions (11R20), Algebraic number theory computations (11Y40).
  • Line 3: Geometry of numbers (11Hxx), Lattices and convex bodies (11H06), Lattice minima and units (11R27), Continued Fractions (11A55),
    Diophantine equations (11Dxx), Quadratic and bilinear equations (11D09), Cubic and quartic equations (11D25).
  • Line 2: Additive number theory (11Pxx), Partitions of integers (05A17), Other additive questions involving primes (11P32),
    Computational number theory (11Yxx), Calculation of integer sequences (11Y55), Analytic theory of prime numbers (11A41).
  • Line 1: Formal power series rings (13F25), Power series rings (13J05), Formal power series algebras with additional structure (16W60),
    Iteration theory (39B12)


Hilbert
[Hi2]
Schreier
[Sr1], [Sr2]
Hilbert
[Hi1]
Kuroda
[Kd]
Hasse
[Ha1]
Markoff
[Mk]
Voronoi
[Vo1],
[Vo2]
Sprague
[Sp]
Breusch
[Bs]
| \ / | | / | | | | |
Lenstra
[Ls]
Artin
[Ar1],
[Ar2]
Hasse
[Ha2],
[Ha3]
Hall
[Hl]
Kubota
[Ku1],
[Ku2]
Honda
[Ho]
Berwick
[Bw]
Gras
[Gr]
Dedekind
[De]
Delone &
Faddeev
[DnFd]
Richert
[Ri]
Molsen
[Ml]
| | \ | | | | | | | \ / | | |
Schoof
[Sf]
Furtwängler
[Fu1],
[Fu2],
[Fu3]
Scholz
[So2]
Blackburn
[Bl]
Couture
& Derhem
[Dh],
[CtDh]
Barrucand
& Williams
& Baniuk
[BWB]
Scholz
[So1]
Gerth
[Ge]
Barrucand
& Cohn
[BaCo1],
[BaCo2]
Wada
[Wa1],
[Wa3]
Brentjes
[Bt]
Lin
[Li]
Rosser &
Schoenfeld
[RoSf]
Chen
[Ch1],
[Ch2]
| | / | / | | / | | | | | | \ | |
Wada
[Wa2]
Scholz &
Taussky
[Ta1],
[SoTa],
[Ta2]
Heider &
Schmithals
[HeSm]
Miech
[Mi]
Ismaïli &
El Mesaoudi
[Is],
[IsMe]
Parry
[Pa]
Reichardt
[Re]
Moser
[Mo]
Halter-Koch
[HK1]
Nakamula
[Na1],
[Na2],
[Na3]
Buchmann
[Bu]
Dressler
& Parker
[DrPk]
Mayer
[Ma2]
Reich &
Schwaiger
[RcSw]
| / | \ | / | | / | | | \ | | | |
Brink
& Gold
[Br],
[BrGo]
Kisilevsky
[Ki1],
[Ki2]
Nebelung
[Ne1],
[Ne2]
Ascione
& Havas &
Leedham-Green
[AHL], [A]
Ayadi
[Ay]
Ennola &
Turunen
[EnTu1],
[EnTu2]
Martinet
& Payan
[Mt],
[MtPn]
Schmithals
[Sm]
Williams
[Wi1],
[Wi2],
[Wi3]
Hellwig
[Hw]
Beach &
Williams
& Zarnke
[BWZ]
Nelson
[Ns]
Mayer
[Ma1]
\ \ / / \ \ / / \ | /
Mayer
[Ma5], [Ma7], [Ma16],
[Ma6], [Ma12], [Ma13], [Ma14], [Ma15],
[DMV2001], [DMV2009]
Mayer
[Ma8], [Ma9], [Ma10], [Ma11]
Mayer
[Ma3], [Ma4]
Line 5 Line 4 Line 3 Line 2 Line 1
/ / \ \ | | |
©1.1 ©2 ©3.1 ©4.1 ® ® ®
| | / \ |
©1.2 ©3.2 ©4.2
|
©3.2.1
|
©3.2.2
|
©3.2.3
|
©3.3



Some of the future developments© announced in the preceding tree diagram,
in particular some Recent Developments of our Research Line 5, have just been completed:




Bibliographical References:

[Ar1] Emil Artin,
Beweis des allgemeinen Reziprozitätsgesetzes,
Abh. Math. Sem. Univ. Hamburg 5 (1927), 353 - 363.

[Ar2] Emil Artin,
Idealklassen in Oberkörpern und allgemeines Reziprozitätsgesetz,
Hambg. Sem.-Abh. 7 (1929), 46 - 51.

[A] J. A. Ascione,
On 3-groups of second maximal class,
Bull. Austral. Math. Soc. 21 (1980), 473 - 474.

[AHL] J. A. Ascione, G. Havas, and C. R. Leedham-Green,
A computer aided classification of certain groups of prime power order,
Bull. Austral. Math. Soc. 17 (1977), 257 - 274, Corrigendum 317 - 319.

[Ay] Mohammed Ayadi,
Sur la capitulation des 3-classes d'idéaux d'un corps cubique cyclique,
Thèse de doctorat, Université Laval, Québec, 1995.

[BaCo1] Pierre Barrucand and Harvey Cohn,
A rational genus, class number divisibility, and unit theory for pure cubic fields,
J. Number Theory 2 (1970), 7 - 21.

[BaCo2] Pierre Barrucand and Harvey Cohn,
Remarks on principal factors in a relative cubic field,
J. Number Theory 3 (1971), 226 - 239.

[BWB] Pierre Barrucand, Hugh C. Williams, and L. Baniuk,
A computational technique for determining the class number of a pure cubic field,
Math. Comp. 30 (1976), no. 134, 312 - 323.

[BWZ] B. D. Beach, Hugh C. Williams and C. R. Zarnke,
Some computer results on units in quadratic and cubic fields,
Proc. 25th Summer Meeting of the Canadian Math. Congress, Lakehead Univ., Thunder Bay, Ontario, 1971, 609 - 648.

[Bw] W. E. H. Berwick,
On cubic fields with a given discriminant,
Proc. London Math. Soc., Ser. 2, 23 (1925), 359-378.

[Bl] Norman Blackburn,
On a special class of p-groups,
Acta Math. 100 (1958), 45 - 92.

[Bs] R. Breusch,
Zur Verallgemeinerung des Bertrandschen Postulates,
daß zwischen x und 2x stets Primzahlen liegen,
Math. Z. 34 (1932), 505 - 526.

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

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

[Bt] A. J. Brentjes,
Multi-dimensional continued fraction algorithms,
Mathematical centre tracts 145, Mathematisch Zentrum, Amsterdam, 1981.

[Bu] Johannes Buchmann,
A generalization of Voronoi's unit algorithm I,
J. Number Theory 20 (1985), 177 - 191.

[Ch1] Kuo-Tsai Chen,
On local diffeomorphisms about an elementary fixed point,
Bull. Amer. Math. Soc. 69 (1963), 838 - 840.

[Ch2] Kuo-Tsai Chen,
Local diffeomorphisms - C realization of formal properties,
Amer. J. Math. 87 (1965), 140 - 157.

[CtDh] Raymond Couture et Aïssa Derhem,
Un problème de capitulation,
C. R. Acad. Sci. Paris, Série I, 314 (1992), 785 - 788.

[De] Richard Dedekind,
Über die Anzahl der Idealklassen in reinen kubischen Zahlkörpern,
J. Reine Angew. Math. 121 (1900), 40 - 123.

[DnFd] B. N. Delone and D. K. Faddeev,
The theory of irrationalities of the third degree,
Trudy Mat. Inst. Steklov. 11 (1940), Transl. Math. Monographs 10, Amer. Math Soc., Providence, Rhode Island, 1964.

[Dh] Aïssa Derhem,
Capitulation dans les extensions quadratiques non ramifiées de corps de nombres cubiques cycliques,
Thèse de doctorat, Université Laval, Québec, 1988.

[DrPk] R. E. Dressler and T. Parker,
"12,758",
Math. Comp., 28 (1974), no. 125, 313 - 314.

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

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

[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.

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

[Gr] Marie-Nicole 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] Philip Hall,
The classification of prime-power groups,
J. reine angew. Math. 182 (1940), 130 - 141.

[HK1] Franz Halter-Koch,
Eine Bemerkung über kubische Einheiten,
Arch. Math. 27 (1976), 593 - 595.

[Ha1] Helmut Hasse,
Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörper. Teil Ia: Beweise zu Teil I,
Jber. der DMV 36 (1927), 233 - 311.

[Ha2] Helmut Hasse,
Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörper. Teil II: Reziprozitätsgesetz,
Jber. der DMV 6 (1930), 1 - 204.

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

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

[Hw] Heino Hellwig,
Eine rationale Interpretation des Klassifikationstheorems von Barrucand und Cohn,
Diplomarbeit, Georg-Augusts-Univ. Göttingen, 1988.

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

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

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

[Is] Moulay Chrif 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.

[IsMe] Moulay Chrif Ismaïli and Rachid 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.

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

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

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

[Ku2] Tomio 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.

[Ls] H. W. Lenstra, Jr.,
On the calculation of regulators and class numbers of quadratic fields
London Math. Soc. Lecture Notes 56 1980, 123 - 150, Journ\'ees arithm\'etiques de Exeter 1980.

[Li] Shen Lin,
Computer experiments on sequences which form integral bases,
pp. 365 - 370 of Computational Problems in Abstract Algebra (J. Leech, editor), Pergamon, Oxford, 1970.

[Mk] A. Markoff,
Sur les nombres entiers dépendants d'une racine cubique d'un nombre entier ordinaire,
Mémoires de l'Académie Impériale des Sciences de St.-Pétersbourg 38 (1892), 7. série, 1 - 37.

[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.

[Ma1] Daniel C. Mayer,
Vektorfelder, Jordan-Zerlegungen und Gruppeneinbettungen formaler Transformationen,
Dissertation, Karl-Franzens-Univ. Graz, Österr. Nationalbibliothek, 1983.

[Ma2] Daniel C. Mayer,
Sharp bounds for the partition function of integer sequences,
BIT 27 (1987), 98 - 110.

[Ma3] Daniel C. Mayer,
Lattice minima and units in real quadratic number fields,
Publicationes Mathematicae Debrecen
39 (1991), 19-86.

[Ma4] Daniel C. Mayer,
Differential principal factors and units in pure cubic number fields,
Dept. of Math., Univ. Graz, 1989.

[Ma5] Daniel C. Mayer,
Multiplicities of dihedral discriminants,
Math. Comp. 58 (1992), no. 198, 831-847, and supplements section S55-S58.

[Ma6] Daniel 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.

[Ma7] Daniel C. Mayer,
Discriminants of metacyclic fields,
Canad. Math. Bull. 36(1) (1993), 103-107.

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

[Ma9] Daniel 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] Daniel C. Mayer,
Principal Factorization Types of Multiplets of Pure Cubic Fields Q( R1/3 ) with R < 106,
Univ. Graz, Computer Centre, 2002.

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

[Ma12] Daniel 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] Daniel 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] Daniel C. Mayer,
Two-Stage Towers of 3-Class Fields over Quadratic Fields,
Univ. Graz, 2006.

[Ma15] Daniel 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] Daniel C. Mayer,
Quadratic p-ring spaces for counting dihedral fields,
Dept. of Computer Science, Univ. of Manitoba, 2009.

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

[Ml] K. Molsen,
Zur Verallgemeinerung des Bertrandschen Postulates,
Deutsche Math. 6 (1941), 248 - 256.

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

[Na1] Ken Nakamula,
Class number calculation and elliptic unit. I. Cubic case,
Proc. Japan Acad. 57 (1981), Ser. A, No. 1, 56 - 59.

[Na2] Ken Nakamula,
Class number calculation of a cubic field from the elliptic unit,
J. Reine Angew. Math. 331 (1982), 114 - 123.

[Na3] Ken Nakamula,
A table for pure cubic fields,
Advanced Studies in Pure Mathematics 13 (1988), Investigations in Number Theory, 461 - 477.

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

[Ne2] Brigitte 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.

[Ns] Harry L. Nelson,
The Partition Problem,
J. Rec. Math. 20 (1988), 315 - 316.

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

[RcSw] L. Reich und J. Schwaiger,
Über einen Satz von Shl. Sternberg in der Theorie der analytischen Iterationen,
Monatsh. Math. 83 (1977), 207-221.

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

[Ri] H.-E. Richert,
Über Zerlegungen in paarweise verschiedene Zahlen,
Norsk Mat. Tidsskr. 31 (1949),120 - 122.

[RoSf] J. B. Rosser and L. Schoenfeld,
Approximate formulas for some functions of prime numbers,
Illinois J. Math. 6 (1962), 64 - 94.

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

[So1] Arnold Scholz,
Über die Beziehung der Klassenzahlen quadratischer Körper zueinander,
J. Reine Angew. Math. 166 (1932), 201-203

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

[SoTa] Arnold Scholz und Olga 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.

[Sf] R. J. Schoof
Quadratic fields and factorization,
Number Theory and Computers, Mathematisch Centrum, Amsterdam, 1980, 235 - 286.

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

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

[Sp] R. Sprague,
Über Zerlegungen in n-te Potenzen mit lauter verschiedenen Grundzahlen,
Math. Z. 51 (1948), 466 - 468.

[Ta1] Olga Taussky,
Über eine Verschärfung des Hauptidealsatzes für algebraische Zahlkörper,
J. Reine Angew. Math. 168 (1932), 193 - 210.

[Ta2] Olga Taussky,
A remark on the class field tower,
J. London Math. Soc. 12 (1937), 82 - 85.

[Vo1] Georgij 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] Georgij F. Voronoi,
Ob odnom obobshchenii algorifma nepreryvnykh drobei
(On a generalization of the algorithm of continued fractions),
Doctoral Dissertation, 1896, Warsaw (Russian).

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

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

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

[Wi1] Hugh C. Williams,
Improving the speed of calculating the regulator of certain pure cubic fields,
Math. Comp. 35 (1980), no. 152, 1423 - 1434.

[Wi2] Hugh C. Williams,
Some results concerning Voronoi's continued fraction over Q(D1/3),
Math. Comp. 36 (1981), no. 154, 631 - 652.

[Wi3] Hugh 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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