| Title:
|
On the Hilbert $2$-class field tower of some abelian $2$-extensions over the field of rational numbers (English) |
| Author:
|
Azizi, Abdelmalek |
| Author:
|
Mouhib, Ali |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
63 |
| Issue:
|
4 |
| Year:
|
2013 |
| Pages:
|
1135-1148 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
It is well known by results of Golod and Shafarevich that the Hilbert $2$-class field tower of any real quadratic number field, in which the discriminant is not a sum of two squares and divisible by eight primes, is infinite. The aim of this article is to extend this result to any real abelian $2$-extension over the field of rational numbers. So using genus theory, units of biquadratic number fields and norm residue symbol, we prove that for every real abelian $2$-extension over $\mathbb Q$ in which eight primes ramify and one of theses primes $\equiv -1\pmod 4$, the Hilbert $2$-class field tower is infinite. (English) |
| Keyword:
|
class group |
| Keyword:
|
class field tower |
| Keyword:
|
multiquadratic number field |
| MSC:
|
11R11 |
| MSC:
|
11R29 |
| MSC:
|
11R37 |
| idZBL:
|
Zbl 06373965 |
| idMR:
|
MR3165518 |
| DOI:
|
10.1007/s10587-013-0075-4 |
| . |
| Date available:
|
2014-01-28T14:25:00Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143620 |
| . |
| Reference:
|
[1] III., F. Gerth: Some real quadratic fields with infinite Hilbert $2$-class field towers.Jap. J. Math., New Ser. 31 (2005), 175-181. Zbl 1075.11066, MR 2153730, 10.4099/math1924.31.175 |
| Reference:
|
[2] Golod, E. S., Shafarevich, I. R.: On the class field tower.Izv. Akad. Nauk SSSR, Ser. Mat. 28 (1964), 261-272 Russian; English translation in Transl., Ser. 2, Am. Math. Soc. 48 (1965), 91-102. MR 0161852 |
| Reference:
|
[3] Hasse, H.: Neue Begründung und Verallgemeinerung der Theorie des Normenrestsymbols.J. f. M. 162 (1930), 134-144 German. |
| Reference:
|
[4] Ishida, M.: The Genus Fields of Algebraic Number Fields. Lecture Notes in Mathematics 555.Springer Berlin (1976). MR 0435028 |
| Reference:
|
[5] Jehne, W.: On knots in algebraic number theory.J. Reine Angew. Math. 311-312 (1979), 215-254. Zbl 0432.12006, MR 0549967 |
| Reference:
|
[6] Kuroda, S.: Über den Dirichletschen Körper.J. Fac. Sci. Univ. Tokyo, Sect. I 4 (1943), 383-406 German. Zbl 0061.05901, MR 0021031 |
| Reference:
|
[7] Kuz'min, L. V.: Homologies of profinite groups, the Schur multiplicator and class field theory.Izv. Akad. Nauk. SSSR Ser. Mat. 33 (1969), 1220-1254 Russian. MR 0255511 |
| Reference:
|
[8] Maire, C.: A refinement of the Golod-Shafarevich theorem. (Un raffinement du théoreme de Golod-Šafarevič).Nagoya Math. J. 150 (1998), 1-11 French. MR 1633138 |
| Reference:
|
[9] Mouhib, A.: On the Hilbert $2$-class field tower of real quadratic fields. (Sur la tour des $2$-corps de classes de Hilbert des corps quadratiques réels).Ann. Sci. Math. Qu. 28 (2004), 179-187 French. MR 2183105 |
| . |
