Franz-Viktor Kuhlmann's

Lecture series on valued function fields and the defect

This is a small course with eight lectures designed for graduate students at the University of Padova.
Here you can find the presentations given in the lectures:

Lecture I: Introduction - local uniformization - ramification theory - elimination of ramification
Lecture II: Lemma of Ostrowski - the defect - examples of defect extensions - defectless fields - tame fields
Lecture III: Valued function fields - valuations on rational function fields - the Abhyankar inequality - Abhyankar valuations - immediate extensions
Lecture IV: The Generalized Stability Theorem and applications - reduction steps in its proof
Lecture V: Proof of the Generalized Stability Theorem: reduction to extensions of prime degree and normal forms
Lecture VI: function fields with non-Abhyankar valuations - alteration - the Henselian Rationality Theorem - first elements of the proof - separable and inseparable local uniformization
Lecture VII: proof of the Henselian Rationality Theorem, normal forms for Galois extensions of degree p, relative approximation degree, reduction steps - embedding theorem for tame fields - application to the model theory of tame fields
Lecture VIII: Zariski spaces of places of function fields - Zariski and patch topology - dense subsets - rational places - large fields.

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Last update: April 23, 2021