I am reading Serre, Local Fields and I am struggling in the proof of this lemma.
The definition the $i$-th ramification group is given by $G_i=\{\sigma\in Gal(L/K):v_L(\sigma a-a)\geq i+1 \forall a\in A_L\}$. It claims that we can take a generator $x$ of $A_L$ as $A_K$-algebra (this is just a uniformizer right?) and that $G_i=\{\sigma\in Gal(L/K):v_L(\sigma x-x)\geq i+1\}$.
The argument he gives is very short, I would very much appreciate if anybody could explain in more detail. I tried proving it by hand, that is taking an element $a\in A_L$ which can be written as $a=\sum a_i x^i$ and then $v_L(\sigma (\sum a_i x_i)-\sum a_ix_i)\geq i+1 \iff v_L(\sigma x-x)\geq i+1$, but this doesn't work.
$A_L=A_K[x]$ implies that $A_L=\sum_{n=0}^{d-1} A_K x^n$ and $v(\sigma (x)-x)\ge i+1$ implies that $$v(\sigma (x^n)-x^n)=v(\sigma (x)-x)+v(\frac{\sigma(x)^n-x^n}{\sigma(x)-x})\ge i+1$$ thus $\forall a\in A_L, a= \sum_{a=0}^{d-1} c_n x^n$ and $v(\sigma (a)-a)\ge \inf_n v(\sigma(c_n x^n)-c_nx^n) \ge i+1$