1. On the Hilbert $2$-class field tower of some abelian $2$-extensions over the field of rational numbers
- Creator:
- Azizi, Abdelmalek and Mouhib, Ali
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- class group, class field tower, and multiquadratic number field
- Language:
- English
- Description:
- 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.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public