Teachers: Yuri Bilu and Denis Benois<\/p>\n\n\n\n
Program:<\/p>\n\n\n\n
1. The field of p-adic numbers: definition and basic properties.
2. A generalization: non-Archimedean complete fields.\u00a0
3. Hensel’s lemma.
4. Extensions of complete fields: ramification, inertia…
5. Applications:<\/p>\n\n\n\n
(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).<\/p>\n\n\n\n
In 5(b) rudiments of p-adic analysis will be studied. In 5(c) the notion of height will be studied.<\/p>\n\n\n\n
Prerequisites:<\/p>\n\n\n\n
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. <\/p>\n\n\n\n
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. <\/p>\n\n\n\n
The course is based on the book <\/p>\n\n\n\n
A. Beshenov, Yu. Bilu, p-adic Numbers and Diophantine Equations, <\/p>\n\n\n\n
currently in press. A printed or electronic version of the book will be made available for the students. <\/p>\n\n\n\n
<\/p>\n","protected":false},"excerpt":{"rendered":"
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.\u00a03. 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 … Continue reading “Number Theory (2026-27)”<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-626","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/uf-mi.u-bordeaux.fr\/algant\/index.php\/wp-json\/wp\/v2\/pages\/626","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/uf-mi.u-bordeaux.fr\/algant\/index.php\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/uf-mi.u-bordeaux.fr\/algant\/index.php\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/uf-mi.u-bordeaux.fr\/algant\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/uf-mi.u-bordeaux.fr\/algant\/index.php\/wp-json\/wp\/v2\/comments?post=626"}],"version-history":[{"count":3,"href":"https:\/\/uf-mi.u-bordeaux.fr\/algant\/index.php\/wp-json\/wp\/v2\/pages\/626\/revisions"}],"predecessor-version":[{"id":641,"href":"https:\/\/uf-mi.u-bordeaux.fr\/algant\/index.php\/wp-json\/wp\/v2\/pages\/626\/revisions\/641"}],"wp:attachment":[{"href":"https:\/\/uf-mi.u-bordeaux.fr\/algant\/index.php\/wp-json\/wp\/v2\/media?parent=626"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}