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Let $L/\Q_p$ be a finite extension, and let $K$ be the maximal unramified subextension. Then $L/K$ is totally ramified, and if $\alpha \in L$ is a uniformizer then the minimal polynomial $\varphi(x) \in K[x]$ of $\alpha$ over $K$ will be Eisenstein. The ramification polygon $P$ of $L$ is the Newton polygon of the ramification polynomial $$\rho(x)=\varphi(\alpha x + \alpha)/(\alpha^n)\in L[x],$$ which is independent of the choice of $\alpha$.

The slopes of the segments of $P$ are the (generalized) lower ramification breaks of $L/K$.

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  • Review status: beta
  • Last edited by David Roe on 2023-03-24 17:35:34
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