Let \(K\) be an algebraic number field of degree \(n=r_ 1+2r_ 2\), with discriminant \(d\) and with ring of integers \(Z_ K\). Denote by \({\mathfrak R}\) the set of integers \(\alpha \in Z_ K\) satisfying \(\alpha\) \(\equiv \gamma \bmod {\mathfrak q}\) (integral ideal), \(Q_ k<\alpha \leq Q_ k+P_ k\) for \(k=1,\ldots,r_ 1\), \(| \alpha^{(k)}| \leq P_ k\) for \(k=r_ 1+1,\ldots,n\). Put \(P=P_ 1\cdot P_ 2\cdots P_ n.\) The main result is an upper estimate for the number of primes \(\omega\in {\mathfrak R}\), for which \(F(\omega)\) is prime, too; here \(F(x)\in Z_ K[X]\) is an irreducible polynomial of degree \(g\geq 1\) with \(F(0)\neq 0\). Denoting by \(\rho(\mathfrak p)\) the number of solutions of \(F(\alpha)\equiv 0 \bmod {\mathfrak p}\) and assuming that \(\rho(\mathfrak p)