## Lecture Notes on p-adic numbers

The following list contains my notes from the p-adic numbers course at UVA this fall, which I’m sitting in on. These notes are taken live during lecture, so beware of any typographical errors!

- Lecture 1: What are the p-adic numbers?
- Lecture 2: Absolute Values on fields
- Lecture 3: Absolute Values (continued)
- Lecture 4: Classification of Absolute Values over $\mathbb{Q}$
- Lecture 5: Completions of $(K,\vert\,\cdot\,\vert)$, Valuations
- Lecture 6: Basic properties of $\mathbb{Q}_p$
- Lecture 7: Some visualizations of $\mathbb{Z}_p$ (to be added)
- Lecture 8: Building towards Hensel’s Lemma
- Lecture 9: Hensel’s Lemma
- Lecture 10: An Effective Hensel’s Lemma
- Lecture 11: Building towards Kummer Congruences
- Lecture 12: Important Gadgets from Fourier Analysis
- Lecture 13: Important Gadgets from Complex Analysis
- Lecture 14: Riemann’s Functional Equation
- Lecture 15: $p$-adic Interpolation, $p$-adic Distributions
- Lecture 16: $p$-adic Distributions (continued)
- Lecture 17: Fixing the Bernoulli Distributions
- Lecture 18: Measure and Integration on the $p$-adics
- Lecture 19: Riemann Integration, more Bernoulli Distributions
- Lecture 20: Proof of Kummer’s Congruence and the Von Staudt-Clausen Theorem
- Lecture 21: Closing Remarks on Kummer; Some Algebraic Number Theory
- Lecture 22: Lifting absolute values to finite extensions