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Let k be an imaginary quadratic number field with C k , 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 Δ k of k, and the number of negative prime discriminants dividing Δ k , 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 Δ k , then k has infinite 2-class field tower.

On imaginary quadratic number fields with 2-class group of rank 4 and infinite 2-class field tower