On Imaginary Quadratic Number Fields with 2-class Group of Rank 4 and Infinite 2-class Field Tower

Let k be an imaginary quadratic number field with Ck,2, the 2-Sylow subgroup of its ideal class group Ck, of rank 4. We show that k has infinite 2-class field tower for particular families of fields k, according to the 4-rank of Ck, the Kronecker symbols of the primes dividing the discriminant ∆k of k, and the number of negative prime discriminants dividing ∆k. In particular we show that if the 4-rank of Ck is greater than or equal to 2 and exactly one negative prime discriminant divides ∆k, then k has infinite 2-class field tower.