### 3rd year group project (10/20cp)

# p-adic numbers

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

### Prerequisites

Algebra 2.

###
References

There are numerous books and online notes on p-adic numbers and local fields.
I think the most accessible books are these two:
J.-P. Serre, A course in arithmetic, GTM 7, Springer, Part I of the book.

Fernando Gouvêa, p-adic numbers: an introduction, Universitext.

These three are also classics, but they are probably more difficult:
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.