Profinite Iterated Monodromy Groups of Unicritical Polynomials

Let $f(x)=a{x}^d+b\in K[x]$ be a unicritical polynomial with degree $d\ge 2$ which is coprime to $\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}K$. We provide an explicit presentation for the profinite iterated monodromy group of $f$, analyze the structure of this group, and use this analysis to determine the constant field extension in $K(f^{-\mathrm{\infty }}(t))\mathrm{/}K(t)$.