Title: Irreducibility and locus of complex roots of polynomials related to Fermat's Last Theorem Show Chinese title

Title: 与费马大定理相关的多项式的不可约性及复根的轨迹

Abstract: We investigate the polynomials $x^n + (1-x)^n + a^n$, a rational root of which would provide a counterexample to Fermat's Last Theorem. We consider the more general question of their irreducibility and prove that in some cases. We investigate the location of complex roots of these polynomials, and prove that for some $a \in \mathbb{Q}$, the roots lie on explicitly given curves while being dense in those curves. Show Chinese abstract

Abstract: 我们研究多项式$x^n + (1-x)^n + a^n$，其有理根将是对费马大定理的反例。 我们考虑它们不可约性的更一般问题，并证明在某些情况下成立。 我们研究这些多项式复根的位置，并证明对于某些$a \in \mathbb{Q}$，根位于明确给定的曲线上，同时在这些曲线上是稠密的。