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p-Adic Numbers
Last Updated: 2026-02-05 15:34:41
Abstract
This course is an introduction to the p-adic numbers. We will see how the field of p-adic numbers Q_p is build. We will explore the (strange) topology and the arithmetic of Q_p, as well as some elementary analytic concepts such as functions, continuity, integrals, etc. We will explain an algebraic and an analytic reasons of interest for the existence of p-adic numbers.
Content
- Absolute values on Q and Completions - Topology and Arithmetic of Q_p, p-adic Integers - Equations over p-adic numbers and Hensel's Lemma - Local-global principle - Hasse-Minkowski's Theorem on binary quadratic forms - Elementary Analysis in Q_p - the p-adic Riemann zeta function
Resources
Literature
"p-adic Numbers. An Introduction", Fernando Q. Gouvea (Springer) "p-adic Numbers, p-adic Analysis, and Zeta-Functions", Neal Koblitz (Springer) "p-adic numbers and Diophantine equations", Yuri Bilu (online notes 2013)
General Information
- Language
- English
- Levels
- BSC , MSC
Examination
- Type
- session examination
- Mode
- written 120 minutes
- Aids
- None.
Course Components
| Type | Title | Time & Place | Hours |
|---|---|---|---|
| lecture |
p-Adic Numbers
"Hybrid"
Students attend classes alternately, a week in presence followed by a week online (except for the first week) according to the scheme described below.
Group 1: family name starting with "A" to "M".
Group 2: family name starting with "N" to "Z".
15.09. Group 1 and Group 2 both online;
22.09., 06.10., 20.10.: Group 1 in presence and Group 2 online;
29.09., 13.10., 27.10.: Group 2 in presence and Group 1 online.
As of November 2020: Group 1 and Group 2 both online.
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2 h weekly |
Offered In
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Electives (For the Master's degree in Applied Mathematics the following additional condition (not manifest in myStudies) must be obeyed: At least 15 of the required 28 credits from core courses and electives must be acquired in areas of applied mathematics and further application-oriented fields.)
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