On the Hilbert 2 -class field tower of some abelian 2 -extensions over the field of rational numbers

Abdelmalek Azizi; Ali Mouhib

Czechoslovak Mathematical Journal (2013)

  • Volume: 63, Issue: 4, page 1135-1148
  • ISSN: 0011-4642

Abstract

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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 ℚ in which eight primes ramify and one of theses primes ≡ - 1 ( mod 4 ) , the Hilbert 2 -class field tower is infinite.

How to cite

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Azizi, Abdelmalek, and Mouhib, Ali. "On the Hilbert $2$-class field tower of some abelian $2$-extensions over the field of rational numbers." Czechoslovak Mathematical Journal 63.4 (2013): 1135-1148. <http://eudml.org/doc/260826>.

@article{Azizi2013,
abstract = {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 $\equiv -1\hspace\{4.44443pt\}(\@mod \; 4)$, the Hilbert $2$-class field tower is infinite.},
author = {Azizi, Abdelmalek, Mouhib, Ali},
journal = {Czechoslovak Mathematical Journal},
keywords = {class group; class field tower; multiquadratic number field; class group; Hilbert class field; class field tower; multiquadratic number field},
language = {eng},
number = {4},
pages = {1135-1148},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the Hilbert $2$-class field tower of some abelian $2$-extensions over the field of rational numbers},
url = {http://eudml.org/doc/260826},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Azizi, Abdelmalek
AU - Mouhib, Ali
TI - On the Hilbert $2$-class field tower of some abelian $2$-extensions over the field of rational numbers
JO - Czechoslovak Mathematical Journal
PY - 2013
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 63
IS - 4
SP - 1135
EP - 1148
AB - 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 $\equiv -1\hspace{4.44443pt}(\@mod \; 4)$, the Hilbert $2$-class field tower is infinite.
LA - eng
KW - class group; class field tower; multiquadratic number field; class group; Hilbert class field; class field tower; multiquadratic number field
UR - http://eudml.org/doc/260826
ER -

References

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  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. (1965) MR0161852
  3. Hasse, H., Neue Begründung und Verallgemeinerung der Theorie des Normenrestsymbols, J. f. M. 162 (1930), 134-144 German. (1930) 
  4. Ishida, M., The Genus Fields of Algebraic Number Fields. Lecture Notes in Mathematics 555, Springer Berlin (1976). (1976) MR0435028
  5. Jehne, W., On knots in algebraic number theory, J. Reine Angew. Math. 311-312 (1979), 215-254. (1979) Zbl0432.12006MR0549967
  6. Kuroda, S., Über den Dirichletschen Körper, J. Fac. Sci. Univ. Tokyo, Sect. I 4 (1943), 383-406 German. (1943) Zbl0061.05901MR0021031
  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. (1969) MR0255511
  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. (1998) MR1633138
  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. (2004) MR2183105

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