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Explore advanced research in valuation theory through this 56-minute mathematical lecture that investigates definable coarsenings of valuation rings and their applications to field extensions. Delve into the theoretical foundations originating from studies of Artin-Schreier and Kummer extensions of valued fields, focusing on the computation of Kähler differentials and associated annihilators. Examine how coarsenings of valuation rings become definable in expanded languages of valued rings, particularly in cases of extensions with independent defect where the maximal ideal equals the unique ramification ideal. Discover the special properties of extensions of prime degree where [L:K]=(vL:vK), including presentations of valuation rings as unions over chains of simple ring extensions and the role of definable maximal ideals in characterizing Kähler differentials. Learn about constructions of deeply ramified fields where arbitrarily assigned convex subgroups of value groups are associated with extensions having independent defect, providing insights into the structural relationships between valuation theory and field extensions in algebraic geometry and commutative algebra.
