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class group; class field tower; multiquadratic number field
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.
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