Sur le rang du $2$-groupe de classes de $Q(\{\sqrt \{m\}\},\{\sqrt \{d\}\})$ où $m=2$ ou un premier $p\equiv 1(mod4)$
HTML articles powered by AMS MathViewer
- by Abdelmalek Azizi and Ali Mouhib PDF
- Trans. Amer. Math. Soc. 353 (2001), 2741-2752 Request permission
Abstract:
On the rank of the $2$-class group of $Q({\sqrt {m}},{\sqrt {d}})$. Let $d$ be a square-free positive integer and $p$ be a prime such that $p\equiv 1 (mod 4)$. We set $K = Q({\sqrt {m}},{\sqrt {d}})$, where $m=2$ or $m=p$. In this paper, we determine the rank of the $2$-class group of $K$.
Résumé. Soit $K = Q({\sqrt {m}},{\sqrt {d}})$, un corps biquadratique où $m=2$ ou bien un premier $p\equiv 1 (mod 4)$ et $d$ étant un entier positif sans facteurs carrés. Dans ce papier, on détermine le rang du $2$-groupe de classes de $K$.
References
- A. Azizi, Capitulation des 2-classes d’idéaux de $Q(\sqrt d;i)$. Thèse. Univ. Laval. Québec. (1993).
- E. Benjamin, F. Lemmermeyer, C. Snyder, Real quadratic fields with abelien 2-class field tower. Research institute of mathematics, Orono, January, 1996.
- I. Benhamza, Unités des corps $Q(\sqrt {d_{1}}, \sqrt {d_{2}},\sqrt {-d})$ et application au problème de capitulation sur le corps $Q(\sqrt {d},\sqrt {-2})$. Thèse. Université Mohamed I. Oujda. (1997).
- Georges Gras, Sur les $l$-classes d’idéaux dans les extensions cycliques relatives de degré premier $l$. I, II, Ann. Inst. Fourier (Grenoble) 23 (1973), no. 3, 1–48; ibid. 23 (1973), no. 4, 1–44 (French, with English summary). MR 360519, DOI 10.5802/aif.471
- Gerald J. Janusz, Algebraic number fields, Pure and Applied Mathematics, Vol. 55, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1973. MR 0366864
- Pierre Kaplan, Divisibilité par $8$ du nombre des classes des corps quadratiques dont le $2$-groupe des classes est cyclique, et réciprocité biquadratique, J. Math. Soc. Japan 25 (1973), 596–608 (French). MR 323757, DOI 10.2969/jmsj/02540596
- Pierre Kaplan, Sur le $2$-groupe des classes des corps quadratiques, Journées Arithmétiques (Grenoble, 1973) Bull. Soc. Math. France Mém. 37, Soc. Math. France, Paris, 1974, pp. 115–116 (French). MR 0399045, DOI 10.24033/msmf.137
- 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
- Radan Kučera, On the parity of the class number of a biquadratic field, J. Number Theory 52 (1995), no. 1, 43–52. MR 1331764, DOI 10.1006/jnth.1995.1054
- S. Kuroda, Über den Dirichletschen Zahlkörper J. Fac. Sc. Imp. Univ. Tokyo Sect. I, 4 (1943). 383-406.
- Thomas M. McCall, Charles J. Parry, and Ramona Ranalli, Imaginary bicyclic biquadratic fields with cyclic $2$-class group, J. Number Theory 53 (1995), no. 1, 88–99. MR 1344833, DOI 10.1006/jnth.1995.1079
- Patrick J. Sime, On the ideal class group of real biquadratic fields, Trans. Amer. Math. Soc. 347 (1995), no. 12, 4855–4876. MR 1333398, DOI 10.1090/S0002-9947-1995-1333398-3
- O. Taussky, A remark on the class field tower. J. London Math. Soc. 12 (1937). 82-85.
- 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
Additional Information
- Abdelmalek Azizi
- Affiliation: Département de Mathématiques, Faculté des Sciences, Université Mohammed 1, Oujda, Maroc
- Email: azizi@sciences.univ-oujda.ac.ma
- Ali Mouhib
- Affiliation: Département de Mathématiques, Faculté des Sciences, Université Mohammed 1, Oujda, Maroc
- Received by editor(s): October 13, 1999
- Received by editor(s) in revised form: July 8, 2000
- Published electronically: February 7, 2001
- © Copyright 2001 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 353 (2001), 2741-2752
- MSC (2000): Primary 11R16, 11R29, 11R37
- DOI: https://doi.org/10.1090/S0002-9947-01-02753-2
- MathSciNet review: 1828471