TITLE: A computation with ramification groups
QUESTION [6 upvotes]: I am trying to solve Exercise 3, Chapter 4, $\S 2$ of Local Fields by Serre.
The exercise is about ramification groups.
Let $L/K$ be a $p$-adic field extension with uniformizer $\pi$. Assume it is Galois with Galois group $G$. Let $e$ be the ramification index of $L$ over $\mathbf Q_p$. Take $s$ in the $i$-th ramification group $G_i$, $i \geq 1$, and denote by $a\in (\pi^i)$ the element such that $s(\pi) = \pi(1+a)$.
Knowing that $s(x)-x \equiv jax \pmod{\pi^{i+j+1}}$ for all $x\in \pi^{j}$, show that for all $x \in (\pi^j)$  and $i > \frac{e}{p-1}$ the following holds:
$$
  s^p(x) -x \equiv pjax \pmod{\pi^{i+j+e+1}}
$$
Does anybody have an idea of how to solve this? The book gives as a hint "use the binomial formula".

REPLY [5 votes]: Write 
$$
s^p-1 = p (s-1) + p(s-1)^2 Q(s-1) + (s-1)^p
$$ 
for some polynomial $Q(T)\in \mathbb Z[T]$, and
notice that 
$$
p(s-1)^2 (x)= 0 \pmod{ \pi^{i+j+e+1}}.
$$