0
votes
0answers
24 views

Question about valuation rings of a rational function field

Let $k$ be a field and $f \in k[x,y] \setminus k$. Write $E=k(f)$ and $L=k(x,y)$. Assume that there exists a $g \in k(x,y)$ such that $L = E(g)$. Suppose there exists a valution ring $\mathcal{O}$ ...
0
votes
0answers
59 views

Application of Hensel's lemma

Show that the polynomial $\Phi(x)=x^2 -2 \in O(\widehat{\Bbb Q_2})[x] $ has no root in $\widehat{\Bbb Q_2}$, even though $\bar\Phi(x)\in E(\widehat{\Bbb Q_2})[x]$ has a root in $E(\widehat{\Bbb Q_2}) ...
2
votes
1answer
126 views

Bounding $p$-adic valuations in inequality

I'm developing an algorithm that comes across inequalities of the form \begin{align*} \operatorname{ord}_p(c(b)) > \alpha \end{align*} for some polynomial $c \in \mathbb{Q}[b]$, $c(b) = c_0 + c_1b + ...