On a converse to the Tschebotarev density theorem

Using an elementary counting procedure on biquadratic polynomials over Zp it is shown that the probability distribution of odd, unramified rational primes according to decomposition type in a fixed dihedral numberfield is identical to the probility of separable quartic polynomials (mod p) whose roots generate numberfields with normal closure having Galois group isomorphic to D4, as p ? 8. This verifies a conjecture about a converse to the Tschebotarev density theorem. Further evidence in support of this conjecture is provided in quadratic and coubic numberfields.