If $f(x)$ is a monic irreducible polynomial in ${\mathbb Z}[x]$ and all roots have absolute value $1$, does that imply $f(x) = x^n+1$?

Question

We know that $ x^n+1$ is a monic polynomial in ${\mathbb Z}[x]$ whose roots are all roots of unity, it is irreducible when $n$ is prime. Are there any other monic irreducible polynomials whose roots all have absolute value $1$ (in particular, this implies all of them are roots of unity ) and not of the form $ x^n+1$?

Answer 1

Any cyclotomic polynomial will have all roots on the unit circle, and not all of those have the form $x^n+1$. For instance, the order-3 cyclotomic polynomial $x^2+x+1$ has all roots on the unit circle (specifically the two non-real cube roots of unity).