To $p$ or not to $p$: Galois' first glimpse of Hensel on wild grounds

In analysing how $p$-henselian valuations can be encoded in the maximal pro-$p$ Galois group of a field it makes a big difference whether the residual characteristic is equal to $p$ or not to $p$. While the first case is well understood, in this talk we will focus on the second presenting first modest results.

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