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Explore computability theory applied to infinite Galois groups in this mathematical lecture that examines how automorphisms of infinite algebraic field extensions can be represented as paths through computable trees. Learn about the Calvert-Harizanov-Shlapentokh framework for describing automorphisms of field extensions E/F and discover how, under basic computability assumptions, these tree structures become computable with Turing-functional composition and inversion operations. Understand the concept of effective presentations for potentially continuum-sized Galois groups and examine how this approach provides remarkably clean results for finite fields' absolute Galois groups. Delve into the notion of "tree-decidability" developed by Block and Miller that formalizes the decidable-like properties of these structures. Contrast the well-behaved finite field case with the more complex absolute Galois group of the rational numbers ℚ, including specific quantifications of this complexity through joint work with Kundu, and consider several open questions about the computational limits and nastiness of these fundamental mathematical objects.
