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Explore the foundational Ax-Kochen/Ershov principles in this mathematical lecture that examines how the first-order theory of henselian valued fields is determined by their residue fields and value groups. Delve into the classical results from sixty years ago showing that complete fields with non-Archimedean absolute values of characteristic zero have theories determined by their component structures. Learn how the original approach transitions from absolute values to valuations and from completeness to henselian conditions to enable model-theoretic methods. Master the key embedding lemma demonstrating how pairs of embeddings between residue fields and value groups extend to valued field embeddings under saturation hypotheses. Discover how this embedding lemma yields Ax-Kochen/Ershov principles for existential theories and extends through back-and-forth arguments to other classical fragments including existential-universal sentences. Examine the state-of-the-art setting of separably tame valued fields and the contributions of Kuhlmann, Knaf, and Pal to the underlying algebraic results. Survey three major extension families: AKE principles for expanded languages including difference and differential fields, analysis of existential theories in henselian valued fields across different characteristics, and groundwork for the Taming Theorem of Jahnke and Kartas addressing valued fields with finite extensions having nontrivial defect.
