RTG Directed Reading: p-adic numbers

Instructor: Alex Mathers
Email: amathers [at] ucsd [dot] edu
Meeting time: Tuesdays 7-8pm PST
Office hours: Fridays 5-6pm PST
Zoom meeting ID: 994 9375 8197

Course description:

In this course we will study the topic of p-adic numbers. In this name p denotes a prime number, so there is a field of p-adic numbers associated to each prime p. Our first goal is to define the p-adic numbers and develop a theory of analysis over them (or more generally for any "nonarchimedean field"), analagous to the theory of analysis over the real or complex numbers. Afterwards we will discuss some algebraic properties of the p-adic numbers and talk about local-global principles.

Course structure:

In lieu of traditional lectures or reading the textbook, I will provide the important material each week in the form of problem sets (link below). Each week we will meet and one student will give a talk summarizing the week's material, which should include some of the important proofs. Giving talks is an important skill for anybody pursuing higher math, and you should take this as an opportunity to improve that skill. I highly recommend that you start the problems as early as possible, especially if you are the week's speaker, as having more time to digest the material will make it easier to summarize the material before your talk.

I will host a Discord server for everybody to communicate while you work through the material. I will also host weekly office hours.

Relevant links:

Course notes: these will be our primary notes for the course.
Ostrowski's Theorem for Q: an exposition of Ostrowski's theorem, from Keith Conrad.
Equivalence of absolute values: an exposition of the equivalent conditions for norms on a field to be equivalent, from Keith Conrad.
Hensel's lemma: an exposition of Hensel's lemma, from Keith conrad.
Pictures of ultrametric spaces, the p-adic numbers, and valued fields: an article describing a visualization of the p-adic numbers, which can lead to intuitive exlanations for some of their properties.
p-adic Numbers: an Introduction, by Fernando Q. Gouvêa: this is the book we will most closely be following, although we will not follow it exactly.
p-adic Analysis Compared with Real, by Svetlana Katok: another book you might consider looking at which develops analysis over the p-adic numbers and makes explicit comparisons to real analysis along the way.
Keith Conrad's blurbs: a collection of expository notes, some about p-adic numbers. There will likely be a few times where I ask that you read a particular article from here.
What does it feel like to invent math?: a video by 3Blue1Brown which introduces some ideas concerning 2-adic numbers and p-adic numbers in general.

Schedule:

Week 1: Definition of normed field, basic notions and topological aspects, definition of the p-adic norm on the rationals.
Week 2: Some properties of nonarchimedean normed fields, sequences, continuity of field operations.
Week 3: More properties in the nonarchimedean and complete cases, constructing the completion, universal property of completion.
Week 4: Ostrowski's theorem, the notion of local-global principles, some algebraic results for the p-adic integers.
Week 5: Description of p-adic integers as inverse limit, p-adic digit expansions, Hensel's lemma.
Week 6: Norms on vector spaces, unique extension of field norms to extensions in the complete case.
Week 7: The p-adic complex numbers, newton polygons.
Week 8: Ramification index and inertia degree, classification of unramified extensions of a complete discretely valued nonarchimedean field.
Week 9: The local Kronecker-Weber theorem.