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Math Help - Algebraic integers ring

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    Algebraic integers ring

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