polynomial with all roots on the unit circle

Question

I'm wondering if the following statement is true: if all roots of a polynomial with integer coefficients are on the unit circle, then these roots are in fact roots of unity and the polynomial is a product of cyclotomic polynomial.

score 14 · Accepted Answer · edited Apr 13 '17 at 12:58

14

This isn't true. Take for example polynomial $5x^2-6x+5$. It's easy to check it has roots $\frac{3}{5}\pm\frac{4}{5}i$, which are both on the unit circle, but neither is a root of unity.

However, if you restrict your attention to monic integer polynomials, then this is indeed correct: it's a result due to Kronecker, and you can see a few proofs of this here.

edited Apr 13 '17 at 12:58

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answered Nov 18 '15 at 16:53

Wojowu

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@NikitaEvseev What do you mean? – Wojowu Nov 18 '15 at 16:59
I thought they will form a regular polygon, but it is wrong – Nikita Evseev Nov 18 '15 at 17:05

polynomial with all roots on the unit circle

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