Real quadratic number fields with metacyclic Hilbert $2$-class field tower

Abstract

We begin by giving a criterion for a number field $K$ with 2-class group of rank 2 to have a metacyclic Hilbert 2-class field tower, and then we will determine all real quadratic number fields $\mathbb Q(\sqrt d)$ that have a metacyclic nonabelian Hilbert $2$-class field tower.

Keywords

• class field tower
• class group
• real quadratic number field
• metacyclic group