Computability Questions About Infinite Galois Groups

Explore computability theory applied to infinite Galois groups through the lens of algebraic field extensions in this mathematical lecture. Discover how automorphisms of infinite algebraic field extensions E/F can be described as paths through computable trees, following the framework established by Calvert, Harizanov, and Shlapentokh. Learn about effective presentations of Galois groups despite their potentially continuum-sized cardinality, and examine how composition and inversion operations remain computable through Turing functionals under basic computability assumptions. Delve into the concept of "tree-decidability" developed by Block and Miller, which provides a rigorous framework for understanding decidable structures in this context. Compare the remarkably well-behaved absolute Galois groups of finite fields with the significantly more complex absolute Galois group of the rational numbers, examining specific quantifications of this complexity including joint work with Kundu. Engage with open questions about the computational limits and decidability properties of these fundamental algebraic structures.