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Explore the foundational principles of tame fields and existential theories through this comprehensive mathematical lecture that delves into the Ax-Kochen-Ershov (AKE) theorem and its modern extensions. Examine how the first-order theory of fields complete with respect to non-Archimedean absolute values in characteristic zero is determined by their residue field and value group theories, building upon the groundbreaking work of Ax, Kochen, and Ershov from sixty years ago. Learn about the transition from absolute values to valuations and from completeness to henselian conditions, which facilitate model-theoretic methods and satisfy Hensel's Lemma conclusions. Discover the embedding lemma for henselian valued fields of equal characteristic zero, where pairs of embeddings between residue fields and value groups extend to valued field embeddings under natural saturation hypotheses. Understand how this lemma enables AKE principles at the existential theory level and extends to other classical fragments through back-and-forth arguments. Investigate the state-of-the-art theory of separably tame valued fields, incorporating principles and algebraic results developed by Kuhlmann and collaborators including Knaf and Pal. Follow the uniform presentation of AKE principles from classical equal characteristic zero settings through extensions to finitely ramified valued fields in mixed characteristic and separably tame fields. Explore three key application families: AKE principles for expanded languages including difference and differential fields, analysis of existential theories in henselian valued fields across different characteristics, and foundational work leading to the Taming Theorem of Jahnke and Kartas for valued fields with finite extensions having nontrivial defect.
