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Let $K$ be a finite extension of $\Q_p$. If we plot points $(n_E,c_E)$ for each intermediate field $\Q_p\subseteq E \subseteq K$, including $K$ and $\Q_p$, where $n_E=[E:\Q_p]$ and $c_E$ its discriminant exponent. Take the lower convex hull of these points.

The visible slopes of $K$ correspond to segments which have slopes greater than one. For such a segment from $(n_1, c_1)$ to $(n_2, c_2)$, $n_2/n_1=p^k$ for some $k\in\Z^+$, and that slope is listed $k$ times.

For a Galois extension $K/\Q_p$, the visible slopes are the same as the wild slopes of the extension. For a general extension, the list of visible slopes is a subset of the list of wild slopes of its Galois closure.

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  • Review status: beta
  • Last edited by John Jones on 2022-05-25 12:23:26
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