Galois groups of some iterated polynomials over cyclotomic extensions

Abstract

Let ${\phi }_p(z)=(z-1{)}^p+2-{\zeta }_p$, where ${\zeta }_p\in \bar{\mathbb{Q}}$ is a primitive $p$-th root of unity for some odd prime $p$. Building on previous work, we show that the $n$-th iterate ${\phi }_p^n(z)$ has Galois group $[C_p{]}^n$, an iterated wreath product of cyclic groups, whenever $p$ is not a Wieferich prime.

arXiv commentWe strengthen the main result to all non-Wieferich primes