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Are all algebraic integers with absolute value 1 roots of unity?

Let $\alpha$ be an algebraic integer. Suppose that all the roots of its minimal polynomial have absolute value 1. Is $\alpha$ a root of unity?

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Makoto Kato
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  • I assume you mean all the roots of its minimal polynomial except possibly $\alpha$, correct? – Alex Becker Jul 25 '12 at 02:18
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    http://math.stackexchange.com/questions/4323/are-all-algebraic-integers-with-absolute-value-1-roots-of-unity?rq=1 – Jack Schmidt Jul 25 '12 at 02:19
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    Dear Makoto, Yes, this is a classical theorem of Kronecker. See @Jack Schmidt's link, and the MO question and answers linked there. Regards, – Matt E Jul 25 '12 at 02:27
  • @MattE Could you provide a link for the MO thread? I don't see one in Jack's link. The question in Jack's link has weaker hypotheses than Makoto's question - Makoto assumes all roots of $\alpha$'s minimal polynomial has absolute value $1$, not just $\alpha$ itself. – Ragib Zaman Jul 25 '12 at 02:31
  • @Ragib: Dear Ragib, See the link to MO in the last comment posted to that question. Regards, – Matt E Jul 25 '12 at 02:37
  • @MattE Ahh, I hadn't expanded to see the hidden comments. Cheers. – Ragib Zaman Jul 25 '12 at 02:38
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    Here is another link http://www2.ucy.ac.cy/~damianou/kronecker.pdf – PAD Jul 25 '12 at 05:53
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    Meta discussion: http://meta.math.stackexchange.com/questions/4745/should-this-question-be-closed-as-duplicate – Willie Wong Jul 25 '12 at 15:36

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