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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.