Do field automorphisms preserve valuation rings? I have a question about valuation fields.  Does an automorphism of  a discrete valuation field necessarily  preserve the valuation ring? (I. e. map elements of absolute value no larger than one to elements with such property.) 
is this the requirement of the definition of the automorphism of DVF?  I guess this should be deduced from the algebraic definition with the help of valuation,  but I don't know how to do this. 
and for more general cases: nondiscrete/archimedean valuations,  what general statements can we obtain?  is there any reference talking about this?  thx! 
 A: For $O_K$ a complete DVR with unique maximal ideal $(\pi_K)$ and a finite extension $L/K$, let $R$ be the integral closure of $O_K$ in $L$, let $\mathfrak{m}$ be a maximal ideal of $R$ (it contains $\pi_K$).
$S = (R-\mathfrak{m})^{-1}R$ is a DVR with uniformizer $\varpi$ that we can choose to be in the intersection of the finitely many maximal ideals of $R$, so that $\varpi^e\in \pi_K R$.
Let $a_1,\ldots,a_f\in R$ be representatives of $S/(\varpi)$. Then $$\varprojlim S/(\varpi^n)$$ is the completion of $S$. In this ring it makes sense to consider $$O_K[a_1,\ldots,a_f][[\varpi]]$$ which is complete with the same uniformizer and residue field, ie. it is the whole of $\varprojlim S/(\varpi^n)$.
But since $\varpi$ is integral over $O_K$ and $\varpi^e\in \pi_K R$ and $O_K$ is complete we have
$$O_K[a_1,\ldots,a_f][[\varpi]]=O_K[a_1,\ldots,a_f,\varpi][[\pi_K]]=O_K[a_1,\ldots,a_f,\varpi]\subset R$$
Thus $$R =O_K[a_1,\ldots,a_f,\varpi]=\varprojlim S/(\varpi^n) \ is \ a\ complete \ DVR$$
Since any automorphism of $L/K$ sends $R$ to itself and since $R$ has only one maximal ideal $(\varpi)$, the automorphisms send the maximal ideal $(\varpi)$ to itself.

It stays true when $O_K$ is Henselian. But when $O_K$ is non-complete nor Henselian it may fail, try with $O_K=(\Bbb{Z}-(5))^{-1}\Bbb{Z},L=\Bbb{Q}(i),R=O_K[i]$, $S=(\Bbb{Z}[i]-(2+i))^{-1}\Bbb{Z}[i]=O_K[i,(2-i)^{-1}],\varpi=(2+i)$ ,$S_2=(\Bbb{Z}[i]-(2-i))^{-1}\Bbb{Z}[i]=O_K[i,(2+i)^{-1}],\varpi_2=2-i$,
the complex conjugaison switch $S$ and $S_2$.

