Imaginary quadratic fields with 2-class group of type (2,2^ℓ)

2013/02/14 by Lopez, Adele
#11R29 11N32 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1302.3453

Abstract

We prove that for any given positive integer ℓ there are infinitely many imaginary quadratic fields with 2-class group of type (2,2^ℓ), and provide a lower bound for the number of such groups with bounded discriminant for ℓ≥2. This work is based on a related result for cyclic 2-class groups by Dominguez, Miller and Wong, and our proof proceeds similarly. Our proof requires introducing congruence conditions into Perelli's result on Goldbach numbers represented by polynomials, which we establitish in some generality.

Related

• Imaginary quadratic fields k with Cl₂(k) = (2,2^m) and Rank Cl₂(k¹) = 2
• Exponents of class groups of certain imaginary quadratic fields
• Imaginary quadratic fields with Cl₂(k) = (2,2,2)
• Quadratic fields with cyclic 2-class groups
• Notes on the divisibility of the class numbers of certain imaginary quadratic fields