proof of some statement about finite field extension I would like to have some explanation for the following statement

Let $K$ be an algebraically closed field of characteristic $p>0$, and $K((t))$, the field of Laurent series with coefficients in $K$. The Galois group of the polynomial $X^{p^n}-X=t^{-1}$ is isomorphic to the additive group of $F_{p^n}$, i.e. to  $(\mathbb{Z}/p\mathbb{Z})^n$.

Another question:
Are there some extensions with Galois group isomorphic to  $(\mathbb{Z}/p^n\mathbb{Z})$ with $n>1$

Say that $\alpha$ is a root of your polynomial $X^{p^n}-X-t^{-1}=0.$ Then it is obvious that if $a \in \mathbb{F}_{p^n},$ that $\alpha+a$ is a root as well, since $(\alpha+a)^{p^n}= \alpha^{p^n}+a^{p^n} = \alpha^{p^n}+a.$ So the Galois group is as claimed.  
There are finite extensions with Galois groups isomorphic to $\mathbb{Z}/p^n\mathbb{Z}$ with $n>1.$ This can be done using the Witt polynomials, see for example: Cyclic Artin-Schreier-Witt extension of order $p^2$ .
