Geodesic
On 2-class field towers for quadratic number fields with 2-class group of type (2,2)
Glasgow mathematical journal, Tome 40 (1998) no. 1, pp. 63-69

Voir la notice de l'article provenant de la source Cambridge University Press

Let K be a quadratic number field with 2-class group of type (2,2). Thus if Sk is the Sylow 2-subgroup of the ideal class group of K, then Sk = Z/2Z × Z/2Z LetK ⊂ K1 ⊂ K2 ⊂ K3 ⊂...the 2-class field tower of K. Thus K1 is the maximal abelian unramified extension of K of degree a power of 2; K2 is the maximal abelian unramified extension of K of 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 of K2 is odd, and hence K2 = K3 = K4 = .... We say that the 2-class field tower of K terminates at K1 if the class number of K1 is odd (and hence K1 = K2 = K3 = ... ); otherwise we say that the 2-class field tower of K terminates at K2. Our goal in this paper is to determine how likely it is for the 2-class field tower of K to terminate at K1 and how likely it is for the 2-class field tower of K to terminate at K2. We shall consider separately the imaginary quadratic fields and the real quadratic fields. 
III, Frank Gerth. On 2-class field towers for quadratic number fields with 2-class group of type (2,2). Glasgow mathematical journal, Tome 40 (1998) no. 1, pp. 63-69. doi: 10.1017/S0017089500032353
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