Teachers: Yuri Bilu and Denis Benois
Program:
1. The field of p-adic numbers: definition and basic properties.
2. A generalization: non-Archimedean complete fields.
3. Hensel’s lemma.
4. Extensions of complete fields: ramification, inertia…
5. Applications:
(a) The Hasse-Minkowski theorem about isotropy of quadratic forms;
(b) The Skolem-Mahler-Lech theorem about vanishing sets of linear recurrent sequences;
(c) Sprindzhuk’s irreducibility theorem (a cute version of Hilbert’s irreducibility theorem).
In 5(b) rudiments of p-adic analysis will be studied. In 5(c) the notion of height will be studied.
Prerequisites:
The students are expected to be familiar with the basic notions of undergraduate algebra and number theory: groups, commutative rings, ideals, linear algebra, finite extensions of fields (including the notions of norm and trace), Galois extensions, congruences (including the Legendre symbol), etc. On a few occasions, we use the resultant of two polynomials and the discriminant of a polynomial.
We will also systematically use the language of general topology: metric spaces, open and closed sets, convergence, compactness, completeness, etc. However, no deep knowledge of topology is expected.
The course is based on the book
A. Beshenov, Yu. Bilu, p-adic Numbers and Diophantine Equations,
currently in press. A printed or electronic version of the book will be made available for the students.