Let $ax^{2}+bxy+cy^{2}$ be a primitive positive definite quadratic form of discriminant $\Delta<0$. It is well known that $ax^{2}+bxy+cy^{2}$ represents infinitely many prime numbers. One of the proofs of this result is elementary and the other, which is more oldest, is based on the Chebotarev density theorem.

My question is if there is other similar results, based on the Chebotarev density theorem, and implying the existence of forms (with degree more than $2$, more variables than $2$) representing infinitely many prime numbers (or other family of numbers).

I would know if there is any other significant results like the theorem above, where the Chebotarev density theorem must be a fundamental ingredient of their proof. I need references.

Thank you for any help.