| Title:
|
On the strongly ambiguous classes of some biquadratic number fields (English) |
| Author:
|
Azizi, Abdelmalek |
| Author:
|
Zekhnini, Abdelkader |
| Author:
|
Taous, Mohammed |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
141 |
| Issue:
|
3 |
| Year:
|
2016 |
| Pages:
|
363-384 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We study the capitulation of \mbox {$2$-ideal} classes of an infinite family of imaginary bicyclic biquadratic number fields consisting of fields $\Bbbk =\Bbb Q(\sqrt {2pq}, {\rm i})$, where ${\rm i}=\sqrt {-1}$ and $p\equiv -q\equiv 1 \pmod 4$ are different primes. For each of the three quadratic extensions $\Bbb K/\Bbbk $ inside the absolute genus field $\Bbbk ^{(*)}$ of $\Bbbk $, we determine a fundamental system of units and then compute the capitulation kernel of $\Bbb K/\Bbbk $. The generators of the groups ${\rm Am}_s(\Bbbk /F)$ and ${\rm Am}(\Bbbk /F)$ are also determined from which we deduce that $\Bbbk ^{(*)}$ is smaller than the relative genus field $(\Bbbk /\Bbb Q({\rm i}))^*$. Then we prove that each strongly ambiguous class of $\Bbbk /\Bbb Q({\rm i})$ capitulates already in $\Bbbk ^{(*)}$, which gives an example generalizing a theorem of Furuya (1977). (English) |
| Keyword:
|
absolute genus field |
| Keyword:
|
relative genus field |
| Keyword:
|
fundamental system of units |
| Keyword:
|
2-class group |
| Keyword:
|
capitulation |
| Keyword:
|
quadratic field |
| Keyword:
|
biquadratic field |
| Keyword:
|
multiquadratic CM-field |
| MSC:
|
11R11 |
| MSC:
|
11R16 |
| MSC:
|
11R20 |
| MSC:
|
11R27 |
| MSC:
|
11R29 |
| MSC:
|
11R37 |
| idZBL:
|
Zbl 06644019 |
| idMR:
|
MR3557585 |
| DOI:
|
10.21136/MB.2016.0022-14 |
| . |
| Date available:
|
2016-10-01T16:03:50Z |
| Last updated:
|
2020-07-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/145899 |
| . |
| Reference:
|
[1] Azizi, A.: On the capitulation of the 2-class group of $\Bbbk=\Bbb Q(\sqrt2pq, i)$ where $p\equiv-q\equiv1\bmod4$.Acta Arith. 94 French (2000), 383-399. MR 1779950 |
| Reference:
|
[2] Azizi, A.: Units of certain imaginary abelian number fields over $\Bbb Q$.Ann. Sci. Math. Qué. 23 French (1999), 15-21. Zbl 1041.11072, MR 1721726 |
| Reference:
|
[3] Azizi, A., Taous, M.: Determination of the fields $\Bbbk=\Bbb Q(\sqrt d,\sqrt{-1})$ given the 2-class groups are of type $(2,4)$ or $(2,2,2)$.Rend. Ist. Mat. Univ. Trieste 40 French (2008), 93-116. MR 2583453 |
| Reference:
|
[4] Azizi, A., Zekhnini, A., Taous, M.: On the strongly ambiguous classes of $\Bbbk/\Bbb Q( i)$ where $\Bbbk=\Bbb Q(\sqrt{2p_1p_2}, i)$.Asian-Eur. J. Math. 7 (2014), Article ID 1450021, 26 pages. MR 3189588 |
| Reference:
|
[5] Azizi, A., Zekhnini, A., Taous, M.: Structure of $ Gal(\Bbbk_2^{(2)}/\Bbbk)$ for some fields $\Bbbk=\Bbb Q(\sqrt{2p_1p_2}, i)$ with $ Cl_2(\Bbbk)\simeq(2,2,2)$.Abh. Math. Semin. Univ. Hamb. 84 (2014), 203-231. MR 3267742 |
| Reference:
|
[6] Azizi, A., Zekhnini, A., Taous, M.: On the generators of the \mbox{$2$-class} group of the field $\Bbbk=\Bbb Q(\sqrt{d}, i)$.Int. J. of Pure and Applied Math. 81 (2012), 773-784. MR 2974674 |
| Reference:
|
[7] Azizi, A., Zekhnini, A., Taous, M.: On the unramified quadratic and biquadratic extensions of the field $\Bbb Q(\sqrt d, i)$.Int. J. Algebra 6 (2012), 1169-1173. Zbl 1284.11142, MR 2974674 |
| Reference:
|
[8] Chevalley, C.: Sur la théorie du corps de classes dans les corps finis et les corps locaux.J. Fac. Sci., Univ. Tokyo, Sect. (1) 2 French (1933), 365-476. Zbl 0008.05301 |
| Reference:
|
[9] Furuya, H.: Principal ideal theorems in the genus field for absolutely Abelian extensions.J. Number Theory 9 (1977), 4-15. Zbl 0347.12006, MR 0429820, 10.1016/0022-314X(77)90045-2 |
| Reference:
|
[10] Gras, G.: Class Field Theory: From Theory to Practice.Springer Monographs in Mathematics Springer, Berlin (2003). Zbl 1019.11032, MR 1941965 |
| Reference:
|
[11] Hasse, H.: On the class number of abelian number fields.Mathematische Lehrbücher und Monographien I Akademie, Berlin German (1952). Zbl 0046.26003, MR 0834791 |
| Reference:
|
[12] Heider, F.-P., Schmithals, B.: Zur Kapitulation der Idealklassen in unverzweigten primzyklischen Erweiterungen.J. Reine Angew. Math. 336 German (1982), 1-25. Zbl 0505.12016, MR 0671319 |
| Reference:
|
[13] Hirabayashi, M., Yoshino, K.-I.: Unit indices of imaginary abelian number fields of type $(2,2,2)$.J. Number Theory 34 (1990), 346-361. Zbl 0705.11065, MR 1049510, 10.1016/0022-314X(90)90141-D |
| Reference:
|
[14] Kubota, T.: Über den bizyklischen biquadratischen Zahlkörper.Nagoya Math. J. 10 German (1956), 65-85. Zbl 0074.03001, MR 0083009, 10.1017/S0027763000000088 |
| Reference:
|
[15] Lemmermeyer, F.: The ambiguous class number formula revisited.J. Ramanujan Math. Soc. 28 (2013), 415-421. MR 3158989 |
| Reference:
|
[16] Louboutin, S.: Hasse unit indices of dihedral octic CM-fields.Math. Nachr. 215 (2000), 107-113. Zbl 0972.11105, MR 1768197, 10.1002/1522-2616(200007)215:1<107::AID-MANA107>3.0.CO;2-A |
| Reference:
|
[17] McCall, T. M., Parry, C. J., Ranalli, R.: Imaginary bicyclic biquadratic fields with cyclic 2-class group.J. Number Theory 53 (1995), 88-99. Zbl 0831.11059, MR 1344833, 10.1006/jnth.1995.1079 |
| Reference:
|
[18] Sime, P. J.: On the ideal class group of real biquadratic fields.Trans. Am. Math. Soc. 347 (1995), 4855-4876. Zbl 0847.11060, MR 1333398, 10.1090/S0002-9947-1995-1333398-3 |
| Reference:
|
[19] Terada, F.: A principal ideal theorem in the genus field.Tohoku Math. J. (2) 23 (1971), 697-718. Zbl 0243.12003, MR 0306158, 10.2748/tmj/1178242555 |
| Reference:
|
[20] Wada, H.: On the class number and the unit group of certain algebraic number fields.J. Fac. Sci., Univ. Tokyo, Sect. (1) 13 (1966), 201-209. Zbl 0158.30103, MR 0214565 |
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