Algebraic integers ring

**tttcomrader** · April 1st 2008, 10:54 AM

Let R be a ring of algebraic integers, and suppose that I is a nonzero ideal of R. Show that I interset $\mathbb {Z} \neq \{ 0 \}$

**ThePerfectHacker** · April 1st 2008, 11:31 AM

Originally Posted by tttcomrader

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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