Let $K=F\_q((X))$, from Artin-Schreier theory, for a cyclic extension $L/K$ of degree p, we have $L=K(x)$, $x^p-x=\alpha$. So when $L/K$ is totally ramified, could we find some $x$ s.t the corresponding $\alpha$ has valuation -1? or $\alpha$ has some bounded valuation independent of $L$? Thanks!



Totally ramified p-extension over $F_q((X))$
Artin-Schreier theory gives a bijection between the set of all degree-$p$ cyclic extension of a field $K$ of prime characteristic $p$ and the set of all $\mathbf{F}\_p$-lines in the $\mathbf{F}\_p$-space $K^+/\wp(K^+)$, where $\wp$ is the endomorphism $x\mapsto x^p-x$ of the additive group $K^+$ (thought of as an $\mathbf{F}\_p$-space).


When $K=k((T))$ for some finite extension $k|\mathbf{F}\_p$ of degree $f=[k:\mathbf{F}\_p]$, the space $K^+/\wp(K^+)$ can be explicitly computed, along with its filtration. This is done for example in [arXiv:0909.2541](http://arxiv.org/abs/0909.2541), from which the following is extracted.


Denote by $\frak{o}$ the ring of integers of $K$, and by $\frak{p}\subset\frak{o}$
the unique maximal ideal of $\frak{o}$. Concretely, we have ${\frak{o}}=k[[T]]$ and
${\frak{p}}=T{\frak{o}}$.


The filtration on the additive group $K^+$ is given by the powers $({\frak{p}}^n)$ of
$\frak{p}$, indexed by $n\in\mathbf{Z}$. It induces a filtration on the $\mathbf{F}\_p$-space $\overline{K^+}=K^+/\wp(K^+)$. We denote the
induced filtration by $\overline{{\frak{p}}^n}$; we have
$\overline{\frak{p}}=\{0\}$, and the codimension at each step is
given by 
$$
\{0\}
\subset\_1\overline{\frak{o}}
\subset\_f\overline{\frak{p}^{-1}}
\cdots
\subset\_f\overline{{\frak{p}}^{-pi+1}}
=\overline{{\frak{p}}^{-pi}}
\subset\_f\cdots
\quad\subset\overline{K^+}
$$
Here, $i$ is any integer $>0$, and an
inclusion of $\mathbf{F}\_p$-spaces $E\subset\_rE'$ means that $E$ is a codimension-$r$
subspace of $E'$.


The unramified degree-$p$ extension of $K$ corresponds to the $\mathbf{F}\_p$-line $\overline{\frak{o}}$ (which can be canonically identified with $\mathbf{F}\_p$ using the trace map $k\to\mathbf{F}\_p$).


So every other line $D$ in this infinite-dimensional $\mathbf{F}\_p$-space $K^+/\wp(K^+)$ corresponds to a ramified cyclic degree-$p$ extension $L=K(\wp^{-1}(D))$ of $K$. If the **level** of $D$ in the displayed filtration is $m$ (say), in the sense that $m>0$ is the smallest integer such that $D\subset\overline{{\frak{p}}^{-m}}$, then the unique ramification break of $\mathrm{Gal}(L|K)$ occurs at $m$.


**Addendum**. On my screen, the \overline does't seem to have any effect, which is very confusing. It is best to read p.17 of the arXiv version referred to above, or 409 of the [published version](https://docs.google.com/open?id=0B8EHtI8F9qdIYWI5YTdmNzUtNmRkMC00NmRiLTliZjAtMTdkNmM0ZjBhMDEy) (*Journal of the Ramanujan Mathematical Society* **25** (2010) 4, 393--417).


