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 C-k, 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 C-k, the Kronecker symbols of the primes dividing the discriminant Deltak of Deltak, and the number of negative prime discriminants dividing Deltak. In particular we show that if the 4-rank of C-k is greater than or equal to 2 and exactly one negative prime discriminant divides Deltak, then k has infinite 2-class field tower.