If $f(x)$ is a monic polynomial in ${\mathbb Z}[x]$ and all roots have absolute value 1 , then all roots are roots of unity.

Question

I am trying to understand the statement that was mentioned here.

If $f(x)$ is a monic polynomial in ${\mathbb Z}[x]$ and all roots have absolute value 1, then all roots are roots of unity.

I was wondering why we need to assume that $f(x)$ is monic. Are there any non-monic polynomials where the conclusion does not hold?

Answer 1

ThePhoenix gives the example $5x^2-6x+5=0$, but $2x^2-3x+2=0$ will do with smaller coefficients. The roots are complex conjugates with the discriminant $-7$, and their product is $2/2=1$ so they must have unit modulus. But the actual roots are given by

$\dfrac{3\pm i\sqrt7}{4},$

and the arguments of these roots are $\pm\cos^{-1}(3/4)$. This can't be a rational multiple of $\pi$ since the cosine is a rational number that isn't half an integer, and roots of unity must have arguments that are rational multiples of $\pi$. $\rightarrow\leftarrow$