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.
Last update: May 23, 2023