 Solution

Choose a nonzero $\alpha \in \mathfrak{a}$. It is an algebraic integer, with monic minimal polynomialIts constant coefficient is nonzero, since otherwise the polynomial would have the root zero and be divisible by its variable, contradicting irreducibility and $\alpha \ne 0$. 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