On the Divisibility of Class Numbers of Cubic Number Fields with   Discriminants in a Prescribed Rational Quadratic Class
  Let n be an odd number and F an imaginary quadratic field with odd discriminant. We show that there exists infinitely many cubic fields K such that the class number of K is divisible by n and the Galois closure of K contains F. 
Ivan Chipchakov and Kalin Kostadinov