On 2-class field towers for quadratic number fields with 2-class group of type (2,2)

Abstract

LetKbe a quadratic number field with 2-class group of type (2,2). Thus ifSkis the Sylow 2-subgroup of the ideal class group ofK, thenSk= ℤ/2ℤ × ℤ/2ℤ LetK ⊂ K1⊂ K2⊂ K3⊂…the 2-class field tower ofK. ThusK1is the maximal abelian unramified extension ofKof degree a power of 2;K2is the maximal abelian unramified extension ofKof degree a power of 2; etc. By class field theory the Galois group Ga1 (K1/K) ≅Sk≅ ℤ/2ℤ × ℤ/2ℤ, and in this case it is known that Ga(K2/Kl) is a cyclic group (cf. [3] and [10]). Then by class field theory the class number ofK2is odd, and henceK2=K3=K4= …. We say that the 2-class field tower ofKterminates atK1if the class number ofK1is odd (and henceK1=K2=K3= … ); otherwise we say that the 2-class field tower ofKterminates atK2. Our goal in this paper is to determine how likely it is for the 2-class field tower ofKto terminate atK1and how likely it is for the 2-class field tower ofKto terminate atK2. We shall consider separately the imaginary quadratic fields and the real quadratic fields.