Inequalities for inert primes and their applications

For any given non-square integer $D\equiv 0,1 \pmod{4}$, we prove Euclid's type inequalities for the sequence $\{q_{i}\}$ of all primes satisfying the Kronecker symbol $(D/q_{i})=-1$, $i=1,2,\cdots,$ and give a new criterion on a ternary quadratic form to be irregular as an application, which simplifies Dickson and Jones's argument in the classification of regular ternary quadratic forms to some extent.