Let R be a ring of algebraic integers, and suppose that I is a nonzero ideal of R. Show that I interset $\displaystyle \mathbb {Z} \neq \{ 0 \} $
Let $\displaystyle \alpha \in R$ be an algebraic integer. Then $\displaystyle \alpha^n + a_{n-1}\alpha^{n-1}+...+a_1\alpha + a_0 = 0$. Where $\displaystyle x^n + a_{n-1}x^{n-1}+...+a_1 x + a_0\in \mathbb{Z}[x]$. It is safe to assume that $\displaystyle a_0\not = 0$ (why?) and so $\displaystyle \alpha^n +a_{n-1}\alpha^{n-1}+...+a_1 \alpha \in I$ because it is an ideal, which means $\displaystyle a_0 \in I$ and so $\displaystyle I\cap \mathbb{Z}\not = \{ 0 \}$.