Primes splits in a quadratic field and inert in a cubic field

Question

Let $F$ be a cubic extension of $\mathbb{Q}$, and let $K$ be a quadratic extension of $\mathbb{Q}$. Is there a positive density of primes $p$ such that $p$ splits in $K$, and $p$ is inert in $F$. I guess the answer is yes, and that the Chebotarev density theorem could apply, but how do we apply it?