This is Proposition 1.9. I am confused with the forward direction:
If $K$ is alg. closed narc field with $x$ a nonzero element with posiive norm. Then:
- $K$ is not discrete.
- Every polynomial over $O_K/(x)$ has a root in $O_K/(x)$.
where $O_K$ is ring of inegers of $K$.
My problems/ thoughts. Suppose I have a polynomial over $O_K/(x)$ of form $\sum a_i t^i$ when lifted to $K$. It has root $y \in K$. But I don't see how we can scale $y$ so as to obtain new root in $O_K/(x)$.
any hints would be appreciated.
The valuation/norm on $K$ is not discrete because $v(x^{n/m}) = v(x) n/m$.
Every monic polynomial $\in O_K[t]$ has a root in $O_K$.
$pt-1\in O_{\overline{Q}_p}/(p^2)[t]$ doesn't have a root in $O_{\overline{Q}_p}/(p^2)$.