Let K be a finite extension over Q_p the fraction field of p-adic integers. Let f(X) =X^{p^r} - c ∈ K[X] where r ∈ Z≥2, and let f_n(X) be the nth iterated polynomial of
f(X). For any a ∈ K, we examine the Galois groups and the ramified index of K_n over K where K_n is the splitting field of f_n(X) − a over K. For some v(c), the behavior
depends on v(c). But for
-p/(p-1) - (r-1)p^r/(p^r-1) ≤ v(c) < -p/(p-1),
we haven’t found results.

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