I'm trying to find an example of a polynomial with some special properties, or an idea of why there can't be one, if that's the case. Here's my setting:
Let $p(x)\in\mathbb{Z}[x]$ be an irreducible, monic polynomial of degree $r$ with complex roots $\alpha_1,\dots,\alpha_r$. A multiplicative relation between its roots is an expression of the form $\alpha_1^{m_1}\dots\alpha_r^{m_r}=1$, where the exponents are integers. This relation is nontrivial if not all exponents are equal.
Clearly, all irreducible reciprocal polynomials (of degree at least 4, to exclude the obvious exception of degree 2) have nontrivial relations between their roots, since these roots appear in pairs of inverses.
Other nontrivial relations, not arising from pairs of inverses, can appear: M. Drmota and M. Skałba give, in their papers "On multiplicative and linear independence of polynomial roots" (1991) and "Relations between polynomial roots" (1995) the example of $x^6-2x^4-6x^3-2x^2+1$, found by A. Schinzel, for which the product of three of its roots is $1$. This relation comes about because the polynomial factors in $\mathbb{Q}(\sqrt{2})$ as $(x^3+\sqrt{2}x^2+\sqrt{2}x-1)(x^3-\sqrt{2}x^2-\sqrt{2}x-1)$. The specific field is not important here: copying this construction but changing $\sqrt{2}$ to $\sqrt{n}$, for non-square $n\in\mathbb{Z}$, we get more examples of irreducible polynomials with nontrivial relations between their roots, and even relations not given by reciprocity. However, all these polynomials are still reciprocal.
What I am looking for, if it exists, is an example of a non-reciprocal polynomial (with integer coefficients, monic and irreducible over $\mathbb{Z}$) with nontrivial relations between its roots. If it does not exists, I would be interested in learing the reason why.
I know that the existence of nontrivial relations limits some aspects of the polynomial: for example, in the first paper I mentioned it is proved that, if an irreducible rational polynomial has prime degree (and is not of the form $x^p+a$), there are no nontrivial relations between its roots. The Galois group of the polynomial is also a factor: there are results showing that if the Galois group of an irreducible rational polynomial of degree $r$ is the full symmetric group $S_r$, there are no nontrivial relations between their roots. The same is true if the Galois group is the alternating group $A_r$. Therefore, maybe a good way to look for an example would be to start by building polynomials with prescribed Galois group. I have tried out some examples of degree $4$ with dihedral Galois group, but without any success.
If anyone could share some insight on this problem, or some references where it is discussed, it would be greatly appreciated!