Corso Valued Fields
Il Prof. Alex Prestel (Univ. Konstanz) sara` ospite del nostro Dipartimento dal 25 Febbraio al 25 Marzo. Durante questo periodo terra` un corso a livello dottorato/laurea specialistica dal titolo VALUED FIELDS
La prima riunione del corso e` fissata Giovedi' 26 Febbraio ore 15 Sala delle riunioni primo piano.
Short Program and Description
Absolute Values Absolute values--Completeness Archimedean complete fields Non-archimedean complete fields
Valuations Ordered abelian groups--Valuations Dependent valuations Construction of valuations Topology-Approximation-Completness
Extension Theory Chevalley--Extension Theorem Algebraic extensions Henselian fields Transcendental extensions
Hilbert Theory Decomposition group--Henselization p-Henselian fields Inertia and ramification group
Two Applications of Valuation Theory Artin--Conjecture A Local-Global-Principle for quadratic forms
Short explanation In Chapter 1 we present the classical theory of absolute values that had its origin in number theory. In Chapter 2 we present Krull's generalization from non-archimedean absolute values to valuations, and deal mainly with the topological aspects of such valuations. In Chapter 3 we present the general extension theory for valuation, and in Chapter 4 we study more carefully algebraic extensions and their relation to certain subgroups of the absolute Galois group of the base field K . In case of a number field K, this theory, applied to non-archimedean absolute values is known as "Hilbert Theory. In Chapter 5, finally we present two applications. We have chosen the applications, so that "general" valuation theory is either necessary for its proof or is essential for its formulation. The applications either deal with number theory or real algebra.
In the first case we sketch a proof of the "almost"-solution of Artin's Conjecture about the p-adic fields Qp . This proof necessarily uses general valuation theory, not covered by the classical theory of absolute values.
In the second case we give a very general Local-Global-Principle for weak isotropy of quadratic forms over formally real fields.
Giulia Curciarello Segreteria Didattica tel: 050-2213219 e-mail curciare@dm.unipi.it _______________________________________________ Settimanale mailing list Settimanale@mail.dm.unipi.it https://mail.dm.unipi.it/mailman/listinfo/settimanale