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55 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}) ...
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120 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 + ...