Geodesic
On the Hilbert $2$-class field tower of some abelian $2$-extensions over the field of rational numbers
Czechoslovak Mathematical Journal, Tome 63 (2013) no. 4, pp. 1135-1148.

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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 $\mathbb Q$ in which eight primes ramify and one of theses primes $\equiv -1\pmod 4$, the Hilbert $2$-class field tower is infinite.
DOI : 10.1007/s10587-013-0075-4
Classification : 11R11, 11R29, 11R37
Keywords: class group; class field tower; multiquadratic number field 
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     title = {On the {Hilbert} $2$-class field tower of some abelian $2$-extensions over the field of rational numbers},
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Azizi, Abdelmalek; Mouhib, Ali. On the Hilbert $2$-class field tower of some abelian $2$-extensions over the field of rational numbers. Czechoslovak Mathematical Journal, Tome 63 (2013) no. 4, pp. 1135-1148. doi : 10.1007/s10587-013-0075-4. http://geodesic.mathdoc.fr/articles/10.1007/s10587-013-0075-4/

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