Bibliographical References:
[Ag]
M. Arrigoni,
On Schur σ-groups,
Math. Nachr.
192
(1998),
71 - 89.
[Ar1]
E. Artin,
Beweis des allgemeinen Reziprozitätsgesetzes,
Abh. Math. Sem. Univ. Hamburg
5
(1927),
353 - 363.
[Ar2]
E. Artin,
Idealklassen in Oberkörpern und allgemeines Reziprozitätsgesetz,
Abh. Math. Sem. Univ. Hamburg
7
(1929),
46 - 51.
[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,
Microfiche Supplement
p.320.
[As1]
J. A. Ascione,
On 3-groups of second maximal class,
Ph.D. Thesis,
Austral. National Univ.,
Canberra,
1979.
[As2]
J. A. Ascione,
On 3-groups of second maximal class,
Bull. Austral. Math. Soc.
21
(1980),
473 - 474.
[Ay] M. Ayadi,
Sur la capitulation des 3-classes d'idéaux d'un corps cubique cyclique,
Thèse de doctorat, Université Laval, Québec, 1995.
[AAI] M. Ayadi, A. Azizi and M. C. Ismaïli
The capitulation problem for certain number fields,
Advanced Studies in Pure Mathemetics,
30
(2001),
Class Field Theory - Its Centenary and Prospect,
pp. 467 - 482.
[Az] A. Azizi,
Sur la capitulation des 2-classes d'idéaux de Q(d1/2,i),
Thèse de doctorat, Université Laval, Québec, 1993,
et C. R. Acad. Sci. Paris, Sér. I,
325
(1997),
127 - 130.
[Az1] A. Azizi,
Sur la capitulation des 2-classes d'idéaux de Q((2pq)1/2,i) où p ≡ -q ≡ 1 mod 4,
Acta Arith.,
94
(2000),
no. 4,
383 - 399.
[AzMh] A. Azizi et A. Mouhib,
Capitulation des 2-classes d'idéaux de Q(21/2,d1/2)
où d est un entier naturel sans facteurs carrés,
Acta Arith. 109 (2003), no. 1, 27 - 63.
[AzMh1] A. Azizi et A. Mouhib,
Sur le 2-groupe de classes du corps de genres de certains corps biquadratiques,
Ann. Sci. Math. Québec 27 (2003), no. 2, 123 - 134.
[AzTb] A. Azizi et M. Talbi,
Capitulation des 2-classes d'idéaux de certains corps biquadratiques cycliques,
Acta Arith. 127 (2007), no. 3, 231 - 248.
[AzTs] A. Azizi et M. Taous,
Capitulation des 2-classes d'idéaux de k=Q((2p)1/2,i),
Acta Arith. 131 (2008), no. 2, 103 - 123.
[AZT] A. Azizi, A. Zekhnini et M. Taous,
Capitulation dans le corps des genres de certain corps de nombres biquadratique imaginaire
dont le 2-groupe des classes est de type (2,2,2),
preprint, 2010.
[Bg]
G. Bagnera,
La composizione dei gruppi finiti
il cui grado è la quinta potenza di un numero primo,
Ann. di Mat.
(Ser. 3)
1
(1898),
137 - 228.
[BaCo1] P. Barrucand and H. Cohn,
A rational genus, class number divisibility, and unit theory for pure cubic fields,
J. Number Theory 2 (1970), 7 - 21.
[BaCo2] P. Barrucand and H. Cohn,
Remarks on principal factors in a relative cubic field,
J. Number Theory 3 (1971), 226 - 239.
[BWB] P. Barrucand, H. 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.
[BtBu]
L. Bartholdi and M. R. Bush,
Maximal unramified 3-extensions of imaginary quadratic fields
and SL2Z3,
J. Number Theory
124
(2007),
159 - 166.
[Be]
K. Belabas,
Topics in computational algebraic number theory,
J. Théor. Nombres Bordeaux
16
(2004),
19 - 63.
[Bb]
T. Bembom,
The capitulation problem in class field theory,
Dissertation,
Georg-August-Universität Göttingen,
2012.
[BjSn]
E. Benjamin and C. Snyder,
Real quadratic number fields with 2-class group of type (2,2),
Math. Scand.
76
(1995),
161 - 178.
[Bw]
W. E. H. Berwick,
On cubic fields with a given discriminant,
Proc. London Math. Soc., Ser. 2, 23 (1925), 359-378.
[BEO]
H. U. Besche, B. Eick, and E. A. O'Brien,
SmallGroups - a library of groups of small order,
2005,
a refereed GAP 4 package.
[Bl1]
N. Blackburn,
On a special class of p-groups,
Acta Math.
100
(1958),
45 - 92.
[Bl2]
N. Blackburn,
On prime-power groups in which the derived group has two generators,
Proc. Camb. Phil. Soc.
53
(1957),
19 - 27.
[Boe]
R. Bölling,
On ranks of class groups of fields in dihedral extensions over Q
with special reference to cubic fields,
Math. Nachr.
135
(1988),
275 - 310.
[BCP]
W. Bosma, J. Cannon, and C. Playoust,
The Magma algebra system. I. The user language,
J. Symbolic Comput.
24
(1997),
235 - 265.
[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.
[ChFt]
S. M. Chang and R. Foote,
Capitulation in class field extensions of type (p,p),
Can. J. Math.
32
(1980),
no. 5,
1229 - 1243.
[CtDh]
R. Couture et A. Derhem,
Un problème de capitulation,
C. R. Acad. Sci. Paris, Série I,
314
(1992),
785 - 788.
[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.
|
|