Irreducibility and locus of complex roots of polynomials related to Fermat's Last Theorem

We study the polynomials [Formula: see text], whose rational roots would yield counterexamples to Fermat’s Last Theorem. We investigate their factorization over [Formula: see text]. In the case [Formula: see text], we ask whether they are irreducible over [Formula: see text], prove the irreducibility for several infinite families, and investigate the location of the roots of these polynomials on the complex plane. For [Formula: see text], the factorization of [Formula: see text] is intimately related to that of the Cauchy–Mirimanoff polynomials [Formula: see text] and the polynomials [Formula: see text] and [Formula: see text] introduced by P. Nanninga. After removing the trivial factors [Formula: see text], [Formula: see text], and [Formula: see text], the remaining components agree (up to change of variable) with [Formula: see text], [Formula: see text], or [Formula: see text]. We prove several new irreducibility results for these factors.