Infinite Hilbert 2-class field tower of quadratic number fields

A. Mouhib

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

Abdelmalek Azizi, Ali Mouhib (2013)

Czechoslovak Mathematical Journal

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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...