Let $K$ be a field complete with respect to a discrete valuation $v$ of residue characteristic $p$. Let $f(z) in K[z]$ be a separable polynomial of the form $z^ell-c.$ Given $a in K$, we examine the Galois groups and ramification groups of the extensions of $K$ generated by the solutions to $f^n(z)=a$. The behavior depends upon $v(c)$, and we find that it shifts dramatically as $v(c)$ crosses a certain value: $0$ in the case $p mid ell$, and $-p/(p-1)$ in the case $p=ell$.
Download