| Title:
|
Real quadratic number fields with metacyclic Hilbert $2$-class field tower (English) |
| Author:
|
Essahel, Said |
| Author:
|
Dakkak, Ahmed |
| Author:
|
Mouhib, Ali |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
144 |
| Issue:
|
2 |
| Year:
|
2019 |
| Pages:
|
177-190 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We begin by giving a criterion for a number field $K$ with 2-class group of rank 2 to have a metacyclic Hilbert 2-class field tower, and then we will determine all real quadratic number fields $\mathbb Q(\sqrt d)$ that have a metacyclic nonabelian Hilbert $2$-class field tower. (English) |
| Keyword:
|
class field tower |
| Keyword:
|
class group |
| Keyword:
|
real quadratic number field |
| Keyword:
|
metacyclic group |
| MSC:
|
11R11 |
| MSC:
|
11R29 |
| MSC:
|
11R37 |
| idZBL:
|
Zbl 07088844 |
| idMR:
|
MR3974186 |
| DOI:
|
10.21136/MB.2018.0102-17 |
| . |
| Date available:
|
2019-06-21T11:33:54Z |
| Last updated:
|
2020-07-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147758 |
| . |
| Reference:
|
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| Reference:
|
[2] Azizi, A., Mouhib, A.: Capitulation of the 2-ideal classes of biquadratic fields whose class field differs from the Hilbert class field.Pac. J. Math. 218 French (2005), 17-36. Zbl 1152.11345, MR 2224587, 10.2140/pjm.2005.218.17 |
| Reference:
|
[3] Benjamin, E., Lemmermeyer, F., Snyder, C.: Real quadratic fields with abelian 2-class field tower.J. Number Theory 73 (1998), 182-194. Zbl 0919.11073, MR 1658015, 10.1006/jnth.1998.2291 |
| Reference:
|
[4] Berkovich, Y., Janko, Z.: On subgroups of finite $p$-group.Isr. J. Math. 171 (2009), 29-49. Zbl 1181.20017, MR 2520099, 10.1007/s11856-009-0038-5 |
| Reference:
|
[5] Martinet, J.: Tours de corps de classes et estimations de discriminants.Invent. Math. 44 French (1978), 65-73. Zbl 0369.12007, MR 0460281, 10.1007/BF01389902 |
| Reference:
|
[6] Mouhib, A.: On the parity of the class number of multiquadratic number fields.J. Number Theory 129 (2009), 1205-1211. Zbl 1167.11039, MR 2521470, 10.1016/j.jnt.2008.12.013 |
| Reference:
|
[7] Mouhib, A.: On 2-class field towers of some real quadratic number fields with 2-class groups of rank 3.Ill. J. Math. 57 (2013), 1009-1018. Zbl 1302.11090, MR 3285864, 10.1215/ijm/1417442559 |
| Reference:
|
[8] Mouhib, A.: A positive proportion of some quadratic number fields with infinite Hilbert 2-class field tower.Ramanujan J. 40 (2016), 405-412. Zbl 06580117, MR 3490564, 10.1007/s11139-015-9713-9 |
| Reference:
|
[9] Taussky, O.: A remark on the class field tower.J. London Math. Soc. 12 (1937), 82-85. Zbl 0016.20002, MR 1574658, 10.1112/jlms/s1-12.1.82 |
| . |
