= Solution
Choose a nonzero $\alpha\in\mathfrak a$. It is an <algebraic integer>, with monic <minimal polynomial>
$$
\alpha^d+c_{d-1}\alpha^{d-1}+\cdots+c_1\alpha+c_0=0,
\qquad c_j\in\mathbb Z.
$$
Its constant coefficient is nonzero, since otherwise the <polynomial> would have the root zero and be divisible by its variable, contradicting irreducibility and $\alpha\ne0$. Rearranging gives $c_0=-\alpha(\alpha^{d-1}+\cdots+c_1)$. The expression in parentheses lies in the <ring of integers of a number field>, so the <ideal> property gives $c_0\in\mathfrak a$. Hence
$$
\boxed{\mathfrak a\cap\mathbb Z\ne\{0\}}.
$$
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