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.

*Last update: May 23, 2023
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