See [Cox, Primes of the form $x^2 + ny^2$], p. 110, Theorem 5.26. There are infinitely many such primes since infinitely many primes split completely in the Hilbert class field, which follows from Chebotarev's density theorem, but there is also a completely elementary argument.
See also [Cox, Primes of the form $x^2 + ny^2$], p. 186, Theorem 9.8(ii).