Article
MSC:
11R11,
11R16,
11R20,
11R27,
11R29,
11R37 | MR 4512171 | Zbl 07655824 | DOI: 10.21136/MB.2022.0056-21
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
multiquadratic field; fundamental systems of units; 2-class group; 2-class field tower; capitulation
multiquadratic field; fundamental systems of units; 2-class group; 2-class field tower; capitulation
Summary:
Let $k$ be a number field with a 2-class group isomorphic to the Klein four-group. The aim of this paper is to give a characterization of capitulation types using group properties. Furthermore, as applications, we determine the structure of the second 2-class groups of some special Dirichlet fields $\Bbbk =\Bbb {Q}\big (\sqrt d, \sqrt {-1}\big )$, which leads to a correction of some parts in the main results of A. Azizi and A. Zekhini (2020).
References:
[1] Azizi, A.: Sur le 2-groupe de classes d'idéaux de $\mathbb Q(\sqrt d, i)$. Rend. Circ. Mat. Palermo, II. Ser. French 48 (1999), 71-92. DOI 10.1007/BF02844380 | MR 1705171 | Zbl 0920.11076
[2] Azizi, A.: Unités de certains corps de nombres imaginaires et abéliens sur $\mathbb Q$. Ann. Sci. Math. Qué. 23 (1999), 15-21 French. MR 1721726 | Zbl 1041.11072
[3] Azizi, A., Benhamza, I.: Sur la capitulation des 2-classes d'idéaux de $\mathbb Q(\sqrt d, \sqrt{-2})$. Ann. Sci. Math. Qué. French 29 (2005), 1-20. MR 2296826 | Zbl 1217.11097
[4] Azizi, A., Zekhnini, A.: The structure of the second 2-class group of some special Dirichlet fields. Asian-Eur. J. Math. 13 (2020), Article ID 2050053, 6 pages. DOI 10.1142/S1793557120500539 | MR 4096421 | Zbl 1442.11154
[5] Azizi, A., Zekhnini, A., Taous, M.: The generators of the 2-class group of some field $\mathbb Q(\sqrt{pq_1q_2,i})$: Correction to Theorem 3 of [5]. Int. J. Pure Appl. Math. 103 (2015), 99-107. DOI 10.12732/ijpam.v103i1.8 | MR 3405725
[6] Chems-Eddin, M. M., Azizi, A., Zekhnini, A.: Unit groups and Iwasawa lambda invariants of some multiquadratic number fields. Bol. Soc. Mat. Mex., III. Ser. 27 (2021), Article ID 24, 16 pages. DOI 10.1007/s40590-021-00329-z | MR 4220815 | Zbl 07342807
[7] Chems-Eddin, M. M., Zekhnini, A., Azizi, A.: Units and 2-class field towers of some multiquadratic number fields. Turk. J. Math. 44 (2020), 1466-1483. DOI 10.3906/mat-2003-117 | MR 4122918 | Zbl 1455.11140
[8] Conner, P. E., Hurrelbrink, J.: Class Number Parity. Series in Pure Mathematics 8. World Scientific, Singapore (1988). DOI 10.1142/0663 | MR 0963648 | Zbl 0743.11061
[9] Gorenstein, D.: Finite Groups. Harper's Series in Modern Mathematics. Harper and Row, New York (1968). MR 0231903 | Zbl 0185.05701
[10] Heider, F.-P., Schmithals, B.: Zur Kapitulation der Idealklassen in unverzweigten primzyklischen Erweiterungen. J. Reine Angew. Math. 366 (1982), 1-25 German. DOI 10.1515/crll.1982.336.1 | MR 0671319 | Zbl 0505.12016
[11] Hilbert, D.: Über den Dirichlet'schen biquadratischen Zahlkörper. Math. Ann. 45 (1894), 309-340 German \99999JFM99999 25.0303.01. DOI 10.1007/BF01446682 | MR 1510866
[12] Kaplan, P.: Sur le 2-groupe des classes d'idéaux des corps quadratiques. J. Reine Angew. Math. 283/284 (1976), 313-363 French. DOI 10.1515/crll.1976.283-284.313 | MR 0404206 | Zbl 0337.12003
[13] Kisilevsky, H.: Number fields with class number congruent to 4 mod 8 and Hilbert's Theorem 94. J. Number Theory 8 (1976), 271-279. DOI 10.1016/0022-314X(76)90004-4 | MR 0417128 | Zbl 0334.12019
[14] Wada, H.: On the class number and the unit group of certain algebraic number fields. J. Fac. Sci., Univ. Tokyo, Sect. I 13 (1966), 201-209. MR 0214565 | Zbl 0158.30103
Search
Partner of
