Greenberg's conjecture and Inverse Galois problem: metacyclic-nonmodular group of type 1 whose abelianization is ℤ/2 ℤ×ℤ/2^m ℤ, m≥ 2

2025/08/25 by Chems-Eddin, Mohamed Mahmoud, Mamry, Hamza El
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2508.18402

Abstract

For an integer m≥ 2, we aim to investigate the realizability of types of metacyclic-nonmodular groups, whose abelianization is ℤ/2 ℤ×ℤ/2^m ℤ, as the Galois group of the maximal unramified 2-extension (resp. pro-2-extension) of certain number fields of 2-power degree (resp. cyclotomic \mathbb Z₂-extensions). Furthermore, we show up some new techniques for studying Greenberg's conjecture for some number fields. In particular, the reader can find results concerning the real quadratic fields F=ℚ(√(ηq rs)), the real biquadratic fields K=ℚ(√(ηq),√(rs)), with η∈\1,2\, and the Fröhlich multiquadratic fields of the form \mathbb F=ℚ(√(q ), √ r, √(s)), where q, r and s are odd primes numbers.

Citations

• Greenberg's conjecture and Iwasawa module of Real biquadratic fields I
• On the unit group and the 2-class number of ℚ(√(2),√(p),√(q))
• Structure of the Galois group of the maximal unramified pro-2-extension of some ℤ₂-extensions
• Arithmetic of some real triquadratic fields; Units and 2-class groups

Related

• On the maximal unramified pro-2-extension of ℤ₂-extension of certain real biquadratic fields
• On Greenberg's generalized conjecture for imaginary quartic fields
• On Greenberg's generalized conjecture for families of number fields
• A weak form of Greenberg's conjecture for the cyclotomic ℤ₂-extension of real quadratic fields
• Greenberg's conjecture and Iwasawa module of Real biquadratic fields I