This is somewhat similar to this recent question, but extending in a different direction.

Let $f(x)$ be an irreducible polynomial of degree $n$ with integer coefficients. Call such $f$ a **bicycle polynomial** if all its roots are located on two circles around $O$, i.e. all roots have one of two moduli. (Of course we'll exclude polynomials of cyclotomic type like $\Phi_n(mx)$ for $m\in\mathbb Z$, which have all their roots fit on *one* circle.)

For $n=2k$ or $n=3k$, examples of bicycle polynomials are $f(x)=g(x^k-1)$, where $g$ is irreducible of degree $2$ or $3$. Taking here a $g$ of degree $4$ with two pairs of complex roots yields still other bicycle polynomials for $n=4k$.

Now: except replacing the "$-1$" by any other nonzero integer, *those types of constructions already seem to be about all of it...*

- Do bicycle polynomials of degree $n$ exist
if $n$ has only prime factors $>3$?

Assuming the existence of such a polynomial, it appears (?) to boil down to the existence of a degree $m$ polynomial ($m>3$ odd) with $m-1$ roots on one circle. This circle must presumably have an irrational radius because of what is known about Salem polynomials, but I'm stuck here.

Also related: How to best distribute points on two concentric circles?

Another question:

- Does anything change if we allow complex integers as coefficients?

Lee-Yang Circle Theorem. See arxiv.org/pdf/1201.3169v1.pdf. Incidentally, the theorem is named after the two physicists who won a Nobel Prize for the overthrow of parity. $\endgroup$ – Richard Stanley Oct 5 '15 at 23:32