Capitulation in certain nonramified extensions of cyclic quartic fields

Abdelmalek Azizi; Mohammed Talbi

Archivum Mathematicum (2008)

  • Volume: 044, Issue: 4, page 271-284
  • ISSN: 0044-8753

Abstract

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Let K = k ( - p ε l ) with k = ℚ ( l ) where l is a prime number such that l = 2 or l ≡ 5 m o d 8 , ε the fundamental unit of k , p a prime number such that p ≡ 1 m o d 4 and ( p l ) 4 = - 1 , K 2 ( 1 ) the Hilbert 2 -class field of K , K 2 ( 2 ) the Hilbert 2 -class field of K 2 ( 1 ) and G = Gal ( K 2 ( 2 ) / K ) the Galois group of K 2 ( 2 ) / K . According to E. Brown and C. J. Parry [7] and [8], C 2 , K , the Sylow 2 -subgroup of the ideal class group of K , is isomorphic to ℤ / 2 ℤ × ℤ / 2 ℤ , consequently K 2 ( 1 ) / K contains three extensions F i / K ( i = 1 , 2 , 3 ) and the tower of the Hilbert 2 -class field of K terminates at either K 2 ( 1 ) or K 2 ( 2 ) . In this work, we are interested in the problem of capitulation of the classes of C 2 , K in F i ( i = 1 , 2 , 3 ) and to determine the structure of G . Résumé. Soient K = k ( - p ε l ) avec k = ℚ ( l ) où l est un nombre premier tel que l = 2 ou l ≡ 5 m o d 8 , ε l’unité fondamentale de k , p un nombre premier tels que p ≡ 1 m o d 4 et ( p l ) 4 = - 1 , K 2 ( 1 ) le 2 -corps de classes de Hilbert de K , K 2 ( 2 ) le 2 -corps de classes de Hilbert de K 2 ( 1 ) et G = Gal ( K 2 ( 2 ) / K ) le groupe de Galois de K 2 ( 2 ) / K . D’après E. Brown et C. J. Parry [7] et [8], C 2 , K , le 2 -groupe de classes de K , est isomorphe à ℤ / 2 ℤ × ℤ / 2 ℤ , par conséquent K 2 ( 1 ) / K contient trois extensions F i / K ( i = 1 , 2 , 3 ) et la tour des 2 -corps de classes de Hilbert de K s’arrête en K 2 ( 1 ) ou en K 2 ( 2 ) . Dans ce travail, on s’intéresse au problème de capitulation des classes de C 2 , K dans F i ( i = 1 , 2 , 3 ) et à déterminer la structure de G .

How to cite

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Azizi, Abdelmalek, and Talbi, Mohammed. "Capitulation dans certaines extensions non ramifiées de corps quartiques cycliques." Archivum Mathematicum 044.4 (2008): 271-284. <http://eudml.org/doc/250477>.

@article{Azizi2008,
author = {Azizi, Abdelmalek, Talbi, Mohammed},
journal = {Archivum Mathematicum},
keywords = {corps biquadratiques cycliques; groupe de classes; capitulation; corps de classes de Hilbert; biquadratic field; capitulation; Hilbert class field},
language = {fre},
number = {4},
pages = {271-284},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Capitulation dans certaines extensions non ramifiées de corps quartiques cycliques},
url = {http://eudml.org/doc/250477},
volume = {044},
year = {2008},
}

TY - JOUR
AU - Azizi, Abdelmalek
AU - Talbi, Mohammed
TI - Capitulation dans certaines extensions non ramifiées de corps quartiques cycliques
JO - Archivum Mathematicum
PY - 2008
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 044
IS - 4
SP - 271
EP - 284
LA - fre
KW - corps biquadratiques cycliques; groupe de classes; capitulation; corps de classes de Hilbert; biquadratic field; capitulation; Hilbert class field
UR - http://eudml.org/doc/250477
ER -

References

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