On the infinitude of 2-class field towers of some imaginary quadratic number fields

By the results of Golod-Shafarevich and Vinberg-Gaschutz, the 2-classfield tower of an imaginary quadratic number field K is infinite if the 2-rankof the ideal class group of K is greater than or equal to 5. In our earlierpapers, we examined the case where the 2-class rank of K is equal to 4, byclassifying the Rìedei matrices of K. We proved that the 2-class field towerof K is infinite if K has only one negative prime discriminant, except for onetype of Rìedei matrix of K, and that the 2-class field tower of K is infinite formany cases if all the prime discriminants of K are negative. In this paper,we investigate the remaining case where K has exactly three negative primediscriminants