Sur la 2-Extension Maximale Non Ramifiée de la Z2-Extension Cyclotomique de Certains Corps Quadratiques

Ali Mouhib

doi:10.2478/auom-2014-0016

.blurhash-client-img { display: none !important; }

Sur la 2-Extension Maximale Non Ramifiée de la Z2-Extension Cyclotomique de Certains Corps Quadratiques

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

Volume 22 (2014): Issue 1 (March 2014)

By: Ali Mouhib

Open Access

|Dec 2014

Abstract

Soient ℓ et ℓ' deux nombres premiers distincts, k = Q(√ℓℓ') et k_∞ la Z₂-extension cyclotomique de k. Soient L_∞ la 2-extension maximale non ramifiée sur k_∞ et L_∞ la sous-extension abélienne maximale de L_∞/k_∞. Sous certaines conditions sur les premiers ℓ et ℓ', nous etudions la structure des groupes de Galois Gal(L_∞/k_∞) et Gal(L_∞/k_∞).

References

[1] A. Azizi, A. Mouhib, Capitulation des 2-classes d'idéaux de Q(√2,√d) oú d est un entier naturel sans facteurs carées, Acta Arith. 109, no 1, 27-73 (2003).
Search in Google Scholar
[2] T. Fukuda, Remarks on Z_p-extensions of number fields, Proc. Japan Acad. Ser. A 70 (1994) 264-266.
Search in Google Scholar
[3] T. Fukuda and K. Komatsu, On the Iwasawa λ-Invariant of the cyclo-tomic Z2-extension of a real quadratic fields, Tokyo J. Math. 28, No. 1 (2005) 259-264.
Search in Google Scholar
[4] B. Ferrero and L. C. Washington, The Iwasawa invariant μ_pvanishes for abelian number fields, Ann. of Math., 109 (1979), 377-395.10.2307/1971116
Search in Google Scholar
[5] R. Greenberg, On the Iwasawa invariants of totally real number fields, Amer. J. Math. 98 (1976) 263-284.10.2307/2373625
Search in Google Scholar
[6] D. Gorenstein, Finite Groups, Harper and Row, new York, 1986.
Search in Google Scholar
[7] K. Iwasawa, On Zl-extensions of algebraic number fields, Ann. of Math. (2) 98 (1973), 246-326.10.2307/1970784
Search in Google Scholar
[8] H. Kisilevsky, Number fields with class number congruent to 4 mod 8 and Hilbert's theorem 94, J. Number Theory 8 (1976) 271-279.10.1016/0022-314X(76)90004-4
Search in Google Scholar
[9] A. Mouhib and A. Movahhedi, p-class tower of a Z_p-extension, Tokyo Journal of Math. vol 31 (2008), 321- 33210.3836/tjm/1233844054
Search in Google Scholar
[10] T. Nguyen Quang Do, M. Lescop, Iwasawa Descent and co-descent for units modulo circular units, Pure and Applied Mathematics Quarterly. Volume 2, Number 2 (2006), 199-217.
Search in Google Scholar
[11] Neukirch, Jürgen; Schmidt, Alexander; Wingberg Kay, Cohomology of number fields, Grundlehren der Mathematischen Wissenschaften [Fundamental principles of Mathematical Sciences], 323. Springer-Verlag, Berlin, 2000.
Search in Google Scholar
[12] M. Ozaki, H. Taya, On the Iwasawa λ₂-invariants of certain families of real quadratic fields. Manuscripta Math. 94 (1997), no. 4, 437-444.
Search in Google Scholar
[13] L. Redei et H. Reichardt, Die Anzahl der durch 4 teilbaren Invarianten der Klassengruppe eines beliebigen quadratischen Zahlkörpers, J. Reine Angew. Math. 170 (1933), 69-74.
Search in Google Scholar