On 2 -class field towers of imaginary quadratic number fields

Franz Lemmermeyer

Journal de théorie des nombres de Bordeaux (1994)

  • Volume: 6, Issue: 2, page 261-272
  • ISSN: 1246-7405

Abstract

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For a number field k , let k 1 denote its Hilbert 2 -class field, and put k 2 = ( k 1 ) 1 . We will determine all imaginary quadratic number fields k such that G = G a l ( k 2 / k ) is abelian or metacyclic, and we will give G in terms of generators and relations.

How to cite

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Lemmermeyer, Franz. "On $2$-class field towers of imaginary quadratic number fields." Journal de théorie des nombres de Bordeaux 6.2 (1994): 261-272. <http://eudml.org/doc/93603>.

@article{Lemmermeyer1994,
abstract = {For a number field $k$, let $k^1$ denote its Hilbert $2$-class field, and put $k^2 = (k^1)^1$. We will determine all imaginary quadratic number fields $k$ such that $G = Gal(k^2/k)$ is abelian or metacyclic, and we will give $G$ in terms of generators and relations.},
author = {Lemmermeyer, Franz},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {Hilbert 2-class field; Galois group; generators; relations},
language = {eng},
number = {2},
pages = {261-272},
publisher = {Université Bordeaux I},
title = {On $2$-class field towers of imaginary quadratic number fields},
url = {http://eudml.org/doc/93603},
volume = {6},
year = {1994},
}

TY - JOUR
AU - Lemmermeyer, Franz
TI - On $2$-class field towers of imaginary quadratic number fields
JO - Journal de théorie des nombres de Bordeaux
PY - 1994
PB - Université Bordeaux I
VL - 6
IS - 2
SP - 261
EP - 272
AB - For a number field $k$, let $k^1$ denote its Hilbert $2$-class field, and put $k^2 = (k^1)^1$. We will determine all imaginary quadratic number fields $k$ such that $G = Gal(k^2/k)$ is abelian or metacyclic, and we will give $G$ in terms of generators and relations.
LA - eng
KW - Hilbert 2-class field; Galois group; generators; relations
UR - http://eudml.org/doc/93603
ER -

References

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  1. [1] E. Benjamin, C. Snyder, Number fields with 2-class groups isomorphic to (2, 2m), Austr. J. Math. 
  2. [2] M. Hall, J.K. Senior, The groups of order 2n(n ≤ 6);, Macmillan, New York (1964). Zbl0192.11701
  3. [3] H. Hasse, Zahlbericht, Physica Verlag, Würzburg, 1965. 
  4. [4] H. Hasse, Über die Klassenzahl abelscher Zahlkörper, Springer Verlag, Heidelberg. Zbl0668.12004
  5. [5] K. Iwasawa, A note on the group of units of an algebraic number field, . Math. pures appl.35 (1956), 189-192. Zbl0071.26504MR76803
  6. [6] P. Kaplan, Sur le 2-groupe des classes d'idéaux des corps quadratiques, J. reine angew. Math.283/284 (1974), 313-363. Zbl0337.12003MR404206
  7. [7] G. Karpilovsky, The Schur multiplier, London Math. Soc. monographs (1987), Oxford. Zbl0619.20001MR1200015
  8. [8] H. Kisilevsky, Number fields with class number congruent to 4 mod 8 and Hilbert's theorem 94, J. Number Theory8 (1976), 271-279. Zbl0334.12019MR417128
  9. [9] H. Koch, Über den 2-Klassenkörperturm eines quadratischen Zahlkörpers, J. reine angew. Math. 214/215 (1963), 201-206. Zbl0123.03904MR164945
  10. [10] F. Lemmermeyer, Die Konstruktion von Klassenkörpern, Diss. Univ. Heidelberg (1994). Zbl0956.11515
  11. [11] L. Rédei, H. Reichardt, Die Anzahl der durch 4 teilbaren Invarianten der Klassengruppe eines beliebigen quadratischen Zahlkörpers, J. reine angew. Math. 170 (1933), 69-74. Zbl0007.39602
  12. [12] A. Scholz, Über die Lösbarkeit der Gleichung t2 - du2 = -4, Math. Z.39 (1934), 95-111. Zbl0009.29402MR1545490JFM60.0126.03
  13. [13] A. Scholz, Abelsche Durchkreuzung, Monatsh. Math. Phys.48 (1939), 340-352. Zbl0023.21101MR623JFM65.0066.02

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