A conjecture of Dedekind asserts that for any finite algebraic extension F of Q, the zeta function $\zeta_F(s)$ is divisible by the Riemann zeta function $\zeta(s)$. That is, the quotient $\zeta_K(s)/\zeta(s)$ is entire. More generally, Dedekind conjectured that if K is a finite extension of F, then $\zeta_K(s)/\zeta_F(s)$ should be entire.
Normally in texts this conjecture is mentioned as a consequence of more general conjectures, like Artin's or Selberg's. But what about the consequences of the Dedekind conjecture itself. For example, we know that the Riemann zeta function gives us much information about the distribution of primes, so I was wondering what (if any) facts about Q or fields in general are known to follow if the Dedekind conjecture is true.