The theory of p-adic numbers is a blend of algebra, number theory and topology, and it is one of the most important tools in modern number theory and algebraic geometry. It has many topics, sidelines and applications that can be introduced more or less independently of one another, and learning it should be well-suited for a group project.
The plan is that we have weekly hour meetings as a group, with students presenting various topics in turn. The precise choice of topics we can decide on later (depending on the students' interests and inclinations), but it could be something along the following lines, in some order:
• p-adic absolute value on Q
• Integers and rationals as p-adic numbers
• Absolute values on fields, equivalence, discrete valuations
• Ostrowski's theorem: classifying absolute values on Q
• Completions of fields and topology
• p-adics vs residue classes mod powers of p
• Units and ring structure
• Application: primitive root theorems
• Equations in p-adics: Hensel's lemma
• Examples: square roots and other power series
• Examples: factoring polynomials
J.-P. Serre, A course in arithmetic, GTM 7, Springer, Part I of the book.
Fernando Gouvêa, p-adic numbers: an introduction, Universitext.
N. Koblitz, p-adic numbers, p-adic analysis, and zeta-functions, GTM 58.
J.-P. Serre, Local fields, GTM 67, Springer.
J.W.S. Cassels, Local fields, CUP.
Questions to Tim Dokchitser.