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

Abstract

We study the polynomials $x^n + (1-x)^n + a^n, a \in\mathbb{Q}$, whose rational roots would yield counterexamples to Fermat's Last Theorem. We investigate their factorization over $\mathbb{Q}$. In the case $a \notin \{0, \pm 1\}$, we ask whether they are irreducible over $\mathbb{Q}$, prove the irreducibility for several infinite families, and investigate the location of the roots of these polynomials on the complex plane. For $a=\pm1$, the factorization of $K_{a,n}$ is intimately related to that of the Cauchy--Mirimanoff polynomials $E_n$ and the polynomials $T_n$ and $S_n$ introduced by P. Nanninga. After removing the trivial factors $x$, $x-1$, and $x^2-x+1$, the remaining components agree (up to change of variable) with $E_n$, $S_n$, or $T_n$. We prove several new irreducibility results for these factors.