| Title:
|
On some imaginary triquadratic number fields $k$ with ${\rm Cl}_2(k) \simeq (2, 4)$ or $(2, 2, 2)$ (English) |
| Author:
|
Azizi, Abdelmalek |
| Author:
|
Chems-Eddin, Mohamed Mahmoud |
| Author:
|
Zekhnini, Abdelkader |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
62 |
| Issue:
|
1 |
| Year:
|
2021 |
| Pages:
|
1-14 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $d$ be a square free integer and $L_d:=\mathbb{Q}(\zeta_{8},\sqrt{d})$. In the present work we determine all the fields $L_d$ such that the $2$-class group, $\mathrm{Cl}_2(L_d)$, of $L_d$ is of type $(2,4)$ or $(2,2,2)$. (English) |
| Keyword:
|
$2$-group rank |
| Keyword:
|
$2$-class group |
| Keyword:
|
imaginary triquadratic number fields |
| MSC:
|
11R11 |
| MSC:
|
11R16 |
| MSC:
|
11R18 |
| MSC:
|
11R27 |
| MSC:
|
11R29 |
| idZBL:
|
Zbl 07396207 |
| idMR:
|
MR4270463 |
| DOI:
|
10.14712/1213-7243.2021.008 |
| . |
| Date available:
|
2021-07-26T11:06:45Z |
| Last updated:
|
2023-04-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/148936 |
| . |
| Reference:
|
[1] Azizi A.: Sur le $2$-groupe de classes d'idéaux de $\mathbb{Q}(\sqrt d, i)$.Rend. Circ. Mat. Palermo (2) 48 (1999), no. 1, 71–92 (French. English summary). MR 1705171, 10.1007/BF02844380 |
| Reference:
|
[2] Azizi A.: Unités de certains corps de nombres imaginaires et abéliens sur $\mathbb{Q}$.Ann. Sci. Math. Québec 23 (1999), no. 1, 15–21 (French. English, French summary). MR 1721726 |
| Reference:
|
[3] Azizi A.: Sur les unités de certains corps de nombres de degré $8$ sur $\mathbb{Q}$.Ann. Sci. Math. Québec 29 (2005), no. 2, 111–129 (French. English, French summary). MR 2309703 |
| Reference:
|
[4] Azizi A., Chems-Eddin M. M., Zekhnini A.: On the rank of the $2$-class group of some imaginary triquadratic number fields.Rend. Circ. Mat. Palermo, II. Ser. (2021), 19 pages. MR 4175725 |
| Reference:
|
[5] Azizi A., Mouhib A.: Capitulation des $2$-classes d'idéaux de $\mathbb{Q}(\sqrt{2},\sqrt{d})$ o\`u $d$ est un entier naturel sans facteurs carrés.Acta. Arith. 109 (2003), no. 1, 27–63 (French). MR 1980850, 10.4064/aa109-1-2 |
| Reference:
|
[6] Azizi A., Taous M.: Capitulation des $2$-classes d'idéaux de $k=\mathbb{Q}(\sqrt{2p},i)$.Acta Arith. 131 (2008), no. 2, 103–123 (French). MR 2388046, 10.4064/aa131-2-1 |
| Reference:
|
[7] Azizi A., Taous M.: Déterminations des corps $K=\mathbb{Q}(\sqrt{d},\sqrt{-1})$ dont les $2$-groupes de classes sont de type $(2, 4)$ ou $(2, 2, 2)$.Rend. Istit. Mat. Univ. Trieste 40 (2008), 93–116 (French. English, French summary). MR 2583453 |
| Reference:
|
[8] Azizi A., Zekhnini A., Taous M.: On the strongly ambiguous classes of $ \mathbbm{k}/\mathbb{Q}(i)$ where $\mathbbm{k}= \mathbb{Q}(\sqrt{2 p_1 p_2},i)$.Asian-Eur. J. Math. 7 (2014), no. 1, 1450021, 26 pages. MR 3189588, 10.1142/S1793557114500211 |
| Reference:
|
[9] Azizi A., Zekhnini A., Taous M.: The generators of the $2$-class group of some fields $\mathbb{Q}(\sqrt{pq_1q_2},i)$: Correction to Theorem $3$ of $[5]$.IJPAM. 103 (2015), no. 1, 99–107. |
| Reference:
|
[10] Azizi A., Zekhnini A., Taous M.: On the unit index of some real biquadratic number fields.Turkish J. Math. 42 (2018), no. 2, 703–715. MR 3794499, 10.3906/mat-1704-63 |
| Reference:
|
[11] Benjamin E., Lemmermeyer F., Snyder C.: Imaginary quadratic fields $k$ with $\mathrm{Cl}_2(k)\simeq (2,2^m)$ and rank $\mathrm{Cl}_2(k^1)=2$.Pacific J. Math. 198 (2001), no. 1, 15–31. MR 1831970 |
| Reference:
|
[12] Benjamin E., Lemmermeyer F., Snyder C.: Imaginary quadratic fields $k$ with $\mathrm{Cl}_2(k)\simeq (2,2,2)$.J. Number Theory 103 (2003), no. 1, 38–70. MR 2008065, 10.1016/S0022-314X(03)00084-2 |
| Reference:
|
[13] Benjamin E., Lemmermeyer F., Snyder C.: On the unit group of some multiquadratic number fields.Pacific J. Math. 230 (2007), no. 1, 27–40. MR 2318446, 10.2140/pjm.2007.230.27 |
| Reference:
|
[14] Brown E.: The class number of $Q(\sqrt{-p})$ for $P\equiv 1(\bmod 8)$ a prime.Proc. Amer. Math. Soc. 31 (1972), 381–383. MR 0289455, 10.1090/S0002-9939-1972-0289455-6 |
| Reference:
|
[15] Conner P. E., Hurrelbrink J.: Class Number Parity.Series in Pure Mathematics, 8, World Scientific Publishing, Singapore, 1988. MR 0963648 |
| Reference:
|
[16] Herglotz G.: Über einen Dirichletschen Satz.Math. Z. 12 (1922), no. 1, 255–261 (German). MR 1544516 |
| Reference:
|
[17] Kaplan P.: 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 0323757 |
| Reference:
|
[18] Kaplan P.: Sur le $2$-groupe des classes d'idéaux des corps quadratiques.J. Reine Angew. Math. 283(284) (1976), 313–363 (French). MR 0404206 |
| Reference:
|
[19] Kučera R.: On the parity of the class number of a biquadratic field.J. Number theory 52 (1995), no. 1, 43–52. MR 1331764, 10.1006/jnth.1995.1054 |
| Reference:
|
[20] Lemmermeyer F.: Ideal class groups of cyclotomic number fields. I.Acta Arith. 72 (1995), no. 4, 347–359. MR 1348202, 10.4064/aa-72-4-347-359 |
| Reference:
|
[21] Kuroda S.: Über die Klassenzahlen algebraischer Zahlkörper.Nagoya Math. J. 1 (1950), 1–10 (German). MR 0039759, 10.1017/S0027763000022777 |
| Reference:
|
[22] Lemmermeyer F.: Kuroda's class number formula.Acta Arith. 66 (1994), no. 3, 245–260. MR 1276992, 10.4064/aa-66-3-245-260 |
| Reference:
|
[23] Leonard P. A., Williams K. S.: On the divisibility of the class numbers of $Q(\sqrt{-p})$ and $Q(\sqrt{-2p})$ by $16$.Canad. Math. Bull. 25 (1982), no. 2, 200–206. MR 0663614, 10.4153/CMB-1982-027-0 |
| Reference:
|
[24] McCall T. M., Parry C. J., Ranalli R.: Imaginary bicyclic biquadratic fields with cyclic $2$-class group.J. Number Theory 53 (1995), no. 1, 88–99. MR 1344833, 10.1006/jnth.1995.1079 |
| Reference:
|
[25] 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 |
| . |
