On the 2-class field tower conjecture for imaginary quadratic number fields with 2-class group of rank 4

Abstract

We demonstrate the existence of infinitely many new imaginary quadratic number fields k with 2-class group Ck,2 of rank 4 such that k has infinite 2-class field tower. In particular, we demonstrate the existence of new fields k as above when the 4-rank of the class group Ck is equal to 1 or 2, and infinitely many new fields k in the case that the 4-rank of Ck is equal to 1, exactly three negative prime discriminants divide the discriminant dk of k, and dk is not congruent to 4 mod 8. This lends support to the conjecture that all imaginary quadratic number fields k with Ck,2 of rank 4 have infinite 2-class field tower.