Local arboreal representations
Jacqueline Anderson, Spencer Hamblen, Bjorn Poonen, and Laura Walton

TL;DR
This paper investigates the Galois groups and ramification structures of iterated polynomial extensions over local fields, revealing how these properties change dramatically depending on valuation thresholds related to the polynomial's parameters.
Contribution
It provides a detailed analysis of how the Galois and ramification groups vary with valuation in local fields for polynomials of the form z^ell - c, highlighting critical valuation thresholds.
Findings
Behavior shifts at valuation zero when p does not divide ell.
Behavior shifts at valuation -p/(p-1) when p equals ell.
Galois group structures depend on the valuation of c.
Abstract
Let be a field complete with respect to a discrete valuation of residue characteristic . Let be a separable polynomial of the form Given , we examine the Galois groups and ramification groups of the extensions of generated by the solutions to . The behavior depends upon , and we find that it shifts dramatically as crosses a certain value: in the case , and in the case .
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