Let $k$ be a local field, and consider the local ring of the formal power series $k[[x_1, \cdots, x_n]]$.
Consider the field of fraction $k((x_1,\cdots, x_n)):=\text{Frac}(K[[x_1, \cdots, x_n]])$.
It is not a local field, but we can define a discrete valuation for it.
I am interested in automorphisms of $k((x_1,\cdots, x_n))$.
- Is there any substantial study in this direction?
For example, if $k=\mathbb Q_p$, then $\text{Aut}(k)=\{id\}$ and therefore the automorphisms of $k((x_1,\cdots, x_n))$ will be determined by the permutations of $x_1, \cdots, x_n$ and in this case, the automorphism will be of infinite order, mostly.
When $k=\mathbb F_p((t))$, in this case also $\text{Aut}(k((x_1,\cdots, x_n)))$ will contain elements of inifinite order.
- How does $\text{Aut}(k)$ extend to $\text{Aut}(k((x_1,\cdots, x_n)))$?
I appreciates on any reference or comments.