Sur une question de capitulation
HTML articles powered by AMS MathViewer
- by Abdelmalek Azizi
- Proc. Amer. Math. Soc. 130 (2002), 2197-2202
- DOI: https://doi.org/10.1090/S0002-9939-02-06424-9
- Published electronically: January 31, 2002
- PDF | Request permission
Abstract:
Let $p$ and $q$ be prime numbers such that $p \equiv 1 \bmod 8, q \equiv -1 \bmod 4$ and $(\frac {\textstyle p}{\textstyle q}) = - 1$. Let $d = pq$, $\mathbf {k} = \mathbf {Q}(\sqrt {d},i)$, and let $\mathbf { k}^{(1)}_{2}$ be the 2-Hilbert class field of $\mathbf {k}$, $\mathbf {k}^{(2)}_{2}$ the 2-Hilbert class field of $\mathbf {k}^{(1)}_{2}$ and $G_{2}$ the Galois group of $\mathbf {k}^{(2)}_{2}/\mathbf {k}$. The 2-part $C_{\mathbf {k},2}$ of the class group of $\mathbf { k}$ is of type $(2,2)$, so $\mathbf {k}_{2}^{(1)}$ contains three extensions $\mathbf {K}_{i}/\mathbf {k}, i = 1, 2, 3$. Our goal is to study the problem of capitulation of the 2-classes of $\mathbf {k}$ in $\mathbf {K}_{i}, i = 1, 2, 3$, and to determine the structure of $G_{2}$.
Résumé. Soient $p$ et $q$ deux nombres premiers tels que $p \equiv 1 \bmod 8, q \equiv -1\bmod 4$ et $(\frac {\textstyle p}{\textstyle q}) = - 1$, $d = pq$, $i = \sqrt {-1}$, $\mathbf {k} = \mathbf {Q}(\sqrt {d},i)$, $\mathbf {k}^{(1)}_{2}$ le 2-corps de classes de Hilbert de $\mathbf {k}$, $\mathbf {k}^{(2)}_{2}$ le 2-corps de classes de Hilbert de $\mathbf {k}^{(1)}_{2}$ et $G_{2}$ le groupe de Galois de $\mathbf {k}^{(2)}_{2}/\mathbf {k}$. La 2-partie $C_{\mathbf {k}, 2}$ du groupe de classes de $\mathbf {k}$ est de type $(2,2)$, par suite $\mathbf {k}^{(1)}_{2}$ contient trois extensions $\mathbf {K}_{i}/\mathbf {k}, i = 1, 2, 3$. On s’intéresse au problème de capitulation des 2-classes de $\mathbf {k}$ dans $\mathbf {K}_{i}, i = 1, 2, 3$, et à déterminer la structure de $G_{2}$.
References
- Abdelmalek Azizi, Sur la capitulation des 2-classes d’idéaux de $\textbf {Q}(\sqrt d,i)$, C. R. Acad. Sci. Paris Sér. I Math. 325 (1997), no. 2, 127–130 (French, with English and French summaries). MR 1467063, DOI 10.1016/S0764-4442(97)84585-5
- Abdelmalek Azizi, Sur le 2-groupe de classes d’idéaux de $\textbf {Q}(\sqrt {d},i)$, Rend. Circ. Mat. Palermo (2) 48 (1999), no. 1, 71–92 (French, with English summary). MR 1705171, DOI 10.1007/BF02844380
- Abdelmalek Azizi, Unités de certains corps de nombres imaginaires et abéliens sur $\mathbf Q$, Ann. Sci. Math. Québec 23 (1999), no. 1, 15–21 (French, with English and French summaries). MR 1721726
- Abdelmalek Azizi, Capitulation of the $2$-ideal classes of $\textbf {Q}(\sqrt {p_1p_2},i)$ where $p_1$ and $p_2$ are primes such that $p_1\equiv 1\pmod 8$, $p_2\equiv 5\pmod 8$ and $(\frac {p_1}{p_2})=-1$, Algebra and number theory (Fez), Lecture Notes in Pure and Appl. Math., vol. 208, Dekker, New York, 2000, pp. 13–19. MR 1724671
- Abdelmalek Azizi, Sur la capitulation des 2-classes d’idéaux de $\textbf {k}=\textbf {Q}(\sqrt {2pq},i)$ où $p\equiv -q\equiv 1\bmod 4$, Acta Arith. 94 (2000), no. 4, 383–399 (French). MR 1779950, DOI 10.4064/aa-94-4-383-399
- Pierre Barrucand and Harvey Cohn, Note on primes of type $x^{2}+32y^{2}$, class number, and residuacity, J. Reine Angew. Math. 238 (1969), 67–70. MR 249396, DOI 10.1515/crll.1969.238.67
- S. M. Chang and R. Foote, Capitulation in class field extensions of type $(p,\,p)$, Canadian J. Math. 32 (1980), no. 5, 1229–1243. MR 596106, DOI 10.4153/CJM-1980-091-9
- Harvey Cohn, The explicit Hilbert $2$-cyclic class field for $\textbf {Q}(\sqrt {-p})$, J. Reine Angew. Math. 321 (1981), 64–77. MR 597980, DOI 10.1515/crll.1981.321.64
- Franz-Peter Heider and Bodo Schmithals, Zur Kapitulation der Idealklassen in unverzweigten primzyklischen Erweiterungen, J. Reine Angew. Math. 336 (1982), 1–25 (German). MR 671319, DOI 10.1515/crll.1982.336.1
- H. Kisilevsky, Number fields with class number congruent to $4$ $\textrm {mod}$ $8$ and Hilbert’s theorem $94$, J. Number Theory 8 (1976), no. 3, 271–279. MR 417128, DOI 10.1016/0022-314X(76)90004-4
- Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1
- 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. MR 1018712
- Hiroshi Suzuki, A generalization of Hilbert’s theorem 94, Nagoya Math. J. 121 (1991), 161–169. MR 1096472, DOI 10.1017/S0027763000003445
- Fumiyuki Terada, A principal ideal theorem in the genus field, Tohoku Math. J. (2) 23 (1971), 697–718. MR 306158, DOI 10.2748/tmj/1178242555
- 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). MR 214565
Bibliographic Information
- Abdelmalek Azizi
- Affiliation: Département de Mathématiques, Faculté des Sciences, Université Mohammed 1, Oujda, Maroc
- Email: azizi@sciences.univ-oujda.ac.ma
- Received by editor(s): February 23, 2001
- Published electronically: January 31, 2002
- Communicated by: David E. Rohrlich
- © Copyright 2002 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 130 (2002), 2197-2202
- MSC (2000): Primary 11R37
- DOI: https://doi.org/10.1090/S0002-9939-02-06424-9
- MathSciNet review: 1897477