An Asymptotic for the Number of Solutions to Linear Equations in Prime Numbers from Specified Chebotarev Classes

We extend known results on the number of solutions to a linear equation in at least three prime numbers when the primes involved are required to lie in specified Chebotarev classes. We prove asymptotic results similar to previous ones only now taking into account corrections coming form the Chebotarev Density Theorem and Global Class Field Theory. We then apply these results to find elliptic curves whose discriminants split completely of a given number field.