Dp-finite fields V: topological fields of finite weight

We prove that unstable dp-finite fields admit definable V-topologies. As a consequence, the henselianity conjecture for dp-finite fields implies the Shelah conjecture for dp-finite fields. This gives a conceptually simpler proof of the classification of dp-finite fields of positive characteristic. For $n \ge 1$, we define a local class of "$W_n$-topological fields", generalizing V-topological fields. A $W_1$-topology is the same thing as a V-topology, and a $W_n$-topology is some higher-rank analogue. If $K$ is an unstable dp-finite field, then the canonical topology is a definable $W_n$-topology for $n = \operatorname{dp-rk}(K)$. Every $W_n$-topology has between 1 and $n$ coarsenings that are V-topologies. If the given $W_n$-topology is definable in some structure, then so are the V-topological coarsenings.

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