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Daniel C. Mayer
Principalization algorithm via class group structure
Journal de théorie des nombres de Bordeaux, 26 no. 2 (2014), p. 415-464, doi: 10.5802/jtnb.874
Article PDF | Reviews MR 3320487
Class. Math.: 11R29, 11R11, 11R16, 11R20, 20D15
Keywords: $3$-class groups, principalization of $3$-classes, quadratic fields, cubic fields, $S_3$-fields, metabelian $3$-groups, coclass graphs

Résumé - Abstract

For an algebraic number field $K$ with $3$-class group $\mathrm{Cl}_3(K)$ of type $(3,3)$, the structure of the $3$-class groups $\mathrm{Cl}_3(N_i)$ of the four unramified cyclic cubic extension fields $N_i$, $1\le i\le 4$, of $K$ is calculated with the aid of presentations for the metabelian Galois group $\mathrm{G}_3^2(K)=\mathrm{Gal}(\mathrm{F}_3^2(K)\vert K)$ of the second Hilbert $3$-class field $\mathrm{F}_3^2(K)$ of $K$. In the case of a quadratic base field $K=\mathbb{Q}(\sqrt{D})$ it is shown that the structure of the $3$-class groups of the four $S_3$-fields $N_1,\ldots ,N_4$ frequently determines the type of principalization of the $3$-class group of $K$ in $N_1,\ldots ,N_4$. This provides an alternative to the classical principalization algorithm by Scholz and Taussky. The new algorithm, which is easily automatizable and executes very quickly, is implemented in PARI/GP and is applied to all $4\,596$ quadratic fields $K$ with $3$-class group of type $(3,3)$ and discriminant $-10^6<D<10^7$ to obtain extensive statistics of their principalization types and the distribution of their second $3$-class groups $\mathrm{G}_3^2(K)$ on various coclass trees of the coclass graphs $\mathcal{G}(3,r)$, $1\le r\le 6$, in the sense of Eick, Leedham-Green, and Newman.

Bibliography

[1] E. Artin, Beweis des allgemeinen Reziprozitätsgesetzes, Abh. Math. Sem. Univ. Hamburg 5, (1927), 353–363.  MR 3069486 |  JFM 53.0144.04
[2] E. Artin, Idealklassen in Oberkörpern und allgemeines Reziprozitätsgesetz. Abh. Math. Sem. Univ. Hamburg 7, (1929), 46–51.  MR 3069515 |  JFM 55.0699.01
[3] J. 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.  MR 470038 |  Zbl 0359.20018
[4] J. Ascione, On $3$-groups of second maximal class, Ph. D. Thesis, Australian National University, Canberra, (1979).
[5] J. Ascione, On $3$-groups of second maximal class. Bull. Austral. Math. Soc., 21, (1980), 473–474.  Zbl 0417.20022
[6] 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.  JFM 29.0112.03
[7] K. Belabas, Topics in computational algebraic number theory. J. Théor. Nombres Bordeaux, 16, (2004), 19–63. Cedram |  MR 2145572 |  Zbl 1078.11071
[8] H. U. Besche, B. Eick, and E. A. O’Brien, The SmallGroups Library — a Library of Groups of Small Order, (2005), an accepted and refereed GAP 4 package, available also in MAGMA.
[9] N. Blackburn, On a special class of $p$-groups, Acta Math., 100, (1958), 45–92.  MR 102558 |  Zbl 0083.24802
[10] N. Blackburn, On prime-power groups in which the derived group has two generators, Proc. Camb. Phil. Soc., 53, (1957), 19–27.  MR 81904 |  Zbl 0077.03202
[11] J. R. Brink, The class field tower for imaginary quadratic number fields of type $(3,3)$, Dissertation, Ohio State University, (1984).
[12] H. Dietrich, B. Eick, and D. Feichtenschlager, Investigating $p$-groups by coclass with GAP. Computational group theory and the theory of groups, Contemp. Math. 470, (2008), 45–61, AMS, Providence, RI.  MR 2478413 |  Zbl 1167.20011
[13] T. E. Easterfield, A classification of groups of order $p^6$, Ph. D. Thesis, University of Cambridge, (1940).
[14] B. Eick and D. Feichtenschlager, Infinite sequences of $p$-groups with fixed coclass, arXiv: 1006.0961 v1 [math.GR], 4 June 2010.
[15] B. Eick and C. Leedham-Green, On the classification of prime-power groups by coclass. Bull. London Math. Soc., 40, (2008), 274–288.  MR 2414786 |  Zbl 1168.20007
[16] 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 $2$, Int. J. Algebra Comput., 23, 5, (2013), 1243–1288.  MR 3096320 |  Zbl 1298.20020
[17] C. Fieker, Computing class fields via the Artin map, Math. Comp., 70, 235 (2001), 1293–1303.  MR 1826583 |  Zbl 0982.11074
[18] G. W. Fung and H. C. Williams, On the computation of a table of complex cubic fields with discriminant $D>-10^6$, Math. Comp., 55, 191, (1990), 313–325.  MR 1023760 |  Zbl 0705.11063
[19] Ph. Furtwängler, Beweis des Hauptidealsatzes für die Klassenkörper algebraischer Zahlkörper, Abh. Math. Sem. Univ. Hamburg, 7, (1929), 14–36.  MR 3069513 |  JFM 55.0699.02
[20] 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.
[21] P. Hall, A contribution to the theory of groups of prime-power order, Proc. London Math. Soc., (ser. 2) 36, (1933), 29–95.  MR 1575964 |  Zbl 0007.29102
[22] P. Hall, The classification of prime-power groups, J. Reine Angew. Math., 182, (1940), 130–141.  MR 3389 |  Zbl 0023.21001
[23] F.-P. Heider und B. Schmithals, Zur Kapitulation der Idealklassen in unverzweigten primzyklischen Erweiterungen, J. Reine Angew. Math., 336, (1982), 1–25.  MR 671319 |  Zbl 0505.12016
[24] R. James, The groups of order $p^6$ ($p$ an odd prime), Math. Comp., 34, 150, (1980), 613–637.  MR 559207 |  Zbl 0428.20013
[25] H. Kisilevsky, Some results related to Hilbert’s theorem $94$, J. Number Theory, 2, (1970), 199–206.  MR 258793 |  Zbl 0216.04701
[26] C. R. Leedham-Green and S. McKay, The structure of groups of prime power order, London Math. Soc. Monographs, New Series, 27, (2002) Oxford Univ. Press.  MR 1918951 |  Zbl 1008.20001
[27] C. R. Leedham-Green and M. F. Newman, Space groups and groups of prime power order I, Arch. Math., 35, (1980), 193–203.  MR 583590 |  Zbl 0437.20016
[28] P. Llorente and J. Quer, On totally real cubic fields with discriminant $D<10^7$, Math. Comp., 50,182, (1988), 581–594.  MR 929555 |  Zbl 0651.12001
[29] D. C. Mayer, Principalization in complex $S_3$-fields, Congressus Numerantium, 80, (1991), 73–87. (Proceedings of the Twentieth Manitoba Conference on Numerical Mathematics and Computing, Winnipeg, Manitoba, Canada, 1990).  MR 1124863 |  Zbl 0733.11037
[30] D. C. Mayer, List of discriminants $d_L<200\,000$ of totally real cubic fields $L$, arranged according to their multiplicities $m$ and conductors $f$, Computer Centre, Department of Computer Science, University of Manitoba, Winnipeg, Canada, (1991).
[31] D. C. Mayer, The second $p$-class group of a number field, Int. J. Number Theory, 8, 2, (2012), 471–505, DOI 10.1142/S179304211250025X.  MR 2890488 |  Zbl 1261.11070
[32] D. C. Mayer, Transfers of metabelian $p$-groups, Monatsh. Math., 166, 3–4, (2012), 467–495, DOI 10.1007/s00605-010-0277-x.  MR 2925150 |  Zbl 1261.11071
[33] D. C. Mayer, Multiplicities of dihedral discriminants, Math. Comp., 58, 198, (1992), 831–847 and S55–S58.  MR 1122071 |  Zbl 0737.11028
[34] D. C. Mayer, The distribution of second $p$-class groups on coclass graphs, J. Théor. Nombres Bordeaux, 25, 2, (2013), 401–456. (27th Journées Arithmétiques, Faculty of Mathematics and Informatics, Vilnius University, Vilnius, Lithuania, 2011). Cedram |  MR 3228314 |  Zbl 1292.11126
[35] D. C. Mayer, Metabelian $3$-groups with abelianization of type $(9,3)$, Preprint, (2011).
[36] R. J. Miech, Metabelian $p$-groups of maximal class, Trans. Amer. Math. Soc., 152, (1970), 331–373.  MR 276343 |  Zbl 0249.20009
[37] B. Nebelung, Klassifikation metabelscher $3$-Gruppen mit Faktorkommutatorgruppe vom Typ $(3,3)$ und Anwendung auf das Kapitulationsproblem, Inauguraldissertation, Band 1, Universität zu Köln, (1989).
[38] B. Nebelung, Anhang zu Klassifikation metabelscher $3$-Gruppen mit Faktorkommutatorgruppe vom Typ $(3,3)$ und Anwendung auf das Kapitulationsproblem, Inauguraldissertation, Band 2, Universität zu Köln, (1989).
[39] The PARI Group, PARI/GP computer algebra system, Version 2.3.4, Bordeaux, (2008), http://pari.math.u-bordeaux.fr.
[40] A. Scholz, Idealklassen und Einheiten in kubischen Körpern, Monatsh. Math. Phys., 40, (1933), 211–222.  MR 1550202 |  Zbl 0007.00301
[41] 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.  Zbl 0009.10202
[42] O. Schreier, Über die Erweiterung von Gruppen II, Abh. Math. Sem. Univ. Hamburg, 4, (1926), 321–346.  MR 3069457 |  JFM 52.0113.04
[43] O. Taussky, A remark concerning Hilbert’s Theorem $94$, J. Reine Angew. Math., 239/240, (1970), 435–438.  MR 279070 |  Zbl 0186.09002
[44] G. F. Voronoĭ, O tselykh algebraicheskikh chislakh zavisyashchikh ot kornya uravneniya tretʼeĭ stepeni, Sanktpeterburg, Master’s Thesis (Russian), (1894). Engl. transl. of title: Concerning algebraic integers derivable from a root of an equation of the third degree.
[45] G. F. Voronoĭ, Ob odnom obobshchenii algorifma nepreryvnykh drobeĭ. Varshava, Doctoral Dissertation (Russian), (1896). Engl. transl. of title: On a generalization of the algorithm of continued fractions, Summary (by Wassilieff): Jahrb. Fortschr. Math. 27, (1896), 170–174.