×

zbMATH — the first resource for mathematics

Sur une question de capitulation. (On the capitulation problem). (French) Zbl 1010.11061
Soient \(p\) et \(q\) deux nombres premiers tels que \(p\equiv 1\pmod 8\), \(q\equiv -1\pmod 4\) et \(({p\over q})=-1\). On considre \(k_2^{(1)}\) le 2-corps de classes de Hilbert de \(k=\mathbb{Q}(\sqrt{pq},\sqrt{-1})\), \(k_2^{(2)}\) le 2-corps de classes de Hilbert de \(k^{(1)}_2\), puis \(G_2\) le groupe de Galois de \(k^{(2) }_2/k\). Comme le 2-groupe de classes de \(k\) est de type \((2,2)\), \(k^{(1)}_2\) contient trois sous-extensions quadratiques \(K_i/k\) \((i=1,2,3)\). A. Azizi étudie alors la capitulation des 2-classes de \(k\) dans \(K_i\) \((i= 1,2, 3)\), et détermine la structure de \(G_2\). Une étude analogue [Acta Arith. 94, 383-399 (2000; Zbl 0953.11033)] avait été menée par le même auteur dans le cas où \(d=2pq\), \(p\equiv -q\equiv 1\pmod 4\), et \(({p\over q})=-1\).

MSC:
11R27 Units and factorization
11R16 Cubic and quartic extensions
11R37 Class field theory
Citations:
Zbl 0953.11033
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Abdelmalek Azizi, Sur la capitulation des 2-classes d’idéaux de \?(\sqrt \?,\?), C. R. Acad. Sci. Paris Sér. I Math. 325 (1997), no. 2, 127 – 130 (French, with English and French summaries). · Zbl 0885.11061
[2] Abdelmalek Azizi, Sur le 2-groupe de classes d’idéaux de \?(\sqrt \?,\?), Rend. Circ. Mat. Palermo (2) 48 (1999), no. 1, 71 – 92 (French, with English summary). · Zbl 0920.11076
[3] Abdelmalek Azizi, Unités de certains corps de nombres imaginaires et abéliens sur \?, Ann. Sci. Math. Québec 23 (1999), no. 1, 15 – 21 (French, with English and French summaries). · Zbl 1041.11072
[4] Abdelmalek Azizi, Capitulation of the 2-ideal classes of \?(\sqrt \?\(_{1}\)\?\(_{2}\),\?) where \?\(_{1}\) and \?\(_{2}\) are primes such that \?\(_{1}\)\equiv 1\pmod8, \?\(_{2}\)\equiv 5\pmod8 and (\frac{\?\(_{1}\)}\?\(_{2}\))=-1, Algebra and number theory (Fez), Lecture Notes in Pure and Appl. Math., vol. 208, Dekker, New York, 2000, pp. 13 – 19. · Zbl 1003.11050
[5] Abdelmalek Azizi, Sur la capitulation des 2-classes d’idéaux de \?=\?(\sqrt 2\?\?,\?) où \?\equiv -\?\equiv 1\bmod4, Acta Arith. 94 (2000), no. 4, 383 – 399 (French). · Zbl 0953.11033
[6] Pierre Barrucand and Harvey Cohn, Note on primes of type \?²+32\?², class number, and residuacity, J. Reine Angew. Math. 238 (1969), 67 – 70. · Zbl 0207.36202
[7] S. M. Chang and R. Foote, Capitulation in class field extensions of type (\?,\?), Canad. J. Math. 32 (1980), no. 5, 1229 – 1243. · Zbl 0459.12007
[8] Harvey Cohn, The explicit Hilbert 2-cyclic class field for \?(\sqrt -\?), J. Reine Angew. Math. 321 (1981), 64 – 77. · Zbl 0455.12006
[9] Franz-Peter Heider and Bodo Schmithals, Zur Kapitulation der Idealklassen in unverzweigten primzyklischen Erweiterungen, J. Reine Angew. Math. 336 (1982), 1 – 25 (German). · Zbl 0505.12016
[10] H. Kisilevsky, Number fields with class number congruent to 4 \?\?\? 8 and Hilbert’s theorem 94, J. Number Theory 8 (1976), no. 3, 271 – 279. · Zbl 0334.12019
[11] T. Kubota, Über den bizyklischen biquadratischen Zahlkörper. Nagoya Math. J, 10 (1956), 65-85. · Zbl 0074.03001
[12] Katsuya Miyake, Algebraic investigations of Hilbert’s Theorem 94, the principal ideal theorem and the capitulation problem, Exposition. Math. 7 (1989), no. 4, 289 – 346. · Zbl 0704.11048
[13] Hiroshi Suzuki, A generalization of Hilbert’s theorem 94, Nagoya Math. J. 121 (1991), 161 – 169. · Zbl 0728.11061
[14] Fumiyuki Terada, A principal ideal theorem in the genus field, Tôhoku Math. J. (2) 23 (1971), 697 – 718. · Zbl 0243.12003
[15] Hideo Wada, On the class number and the unit group of certain algebraic number fields, J. Fac. Sci. Univ. Tokyo Sect. I 13 (1966), 201 – 209 (1966). · Zbl 0158.30103
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.