Communications of the
Korean Mathematical Society CKMS

For an irreducible binomial polynomial $f(x)=x^n-b \in K[x]$ with a field $K$, we ask when does the $m$th iteration $f_m$ is irreducible but $m+1$th $f_{m+1}$ is reducible over $K$. Let $S(n,m)$ be the set of $b$'s such that $f_m$ is irreducible but $f_{m+1}$ is reducible over $K$. We investigate the set $S(n,m)$ by taking $K$ as the rational number field.

Keywords: iterated polynomial, Diophantine equation, ABC conjecture

MSC numbers: 12E05, 11D41