Irreducibility and locus of complex roots of polynomials related to Fermat's Last Theorem
Hayk Karapetyan, Ruben Hambardzumyan
TL;DR
This work investigates polynomials $K_{a,n}(x)=x^n+(1-x)^n+a^n$ in relation to Fermat’s Last Theorem, proving irreducibility for many cases when $a\neq\pm1$ and establishing square-freeness and non-unit-circle roots. For $a=\pm1$, the study connects $K_{a,n}$ to Cauchy–Mirimanoff polynomials $E_n$, $T_n$, $S_n$ and introduces $\tilde{K}_n$ to capture their irreducibility structure, including a detailed root-localization picture on curves and a symmetry framework. The authors develop infinite families of irreducible $\tilde{K}_{2m}$ via Eisenstein and modular methods, analyze the action of a dihedral $H\cong S_3$ on roots, and derive discriminant formulas that yield information on Galois groups. They show, for many $n$ with $n\not\equiv 4\pmod{12}$, the Galois group of $\tilde{K}_n$ contains odd permutations and provide a structural description as an extension of subgroups of wreath-product type, while identifying a remaining open case $n=12m+4$ for future work.
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.