- construction of p-adic numbers,
- p-adic integers, units,
- p-adic topology,
- Hensel's lemma,
- algebraic closure,
- Newton polygon,
- inertia and ramification groups
Module MAT-40-24-M-4
p-adic Numbers (M, 4.5 LP)
Module Identification
Module Number | Module Name | CP (Effort) |
---|---|---|
MAT-40-24-M-4 | p-adic Numbers | 4.5 CP (135 h) |
Basedata
CP, Effort | 4.5 CP = 135 h |
---|---|
Position of the semester | 1 Sem. irreg. SuSe |
Level | [4] Bachelor (Specialization) |
Language | [EN] English |
Module Manager | |
Lecturers | |
Area of study | [MAT-AGCA] Algebra, Geometry and Computer Algebra |
Reference course of study | [MAT-88.105-SG] M.Sc. Mathematics |
Livecycle-State | [NORM] Active |
Courses
Type/SWS | Course Number | Title | Choice in Module-Part | Presence-Time / Self-Study | SL | SL is required for exa. | PL | CP | Sem. | |
---|---|---|---|---|---|---|---|---|---|---|
2V+1U | MAT-40-24-K-4 | p-adic Numbers
| P | 42 h | 93 h |
U-Schein
| - | PL1 | 4.5 | irreg. SuSe |
- About [MAT-40-24-K-4]: Title: "p-adic Numbers"; Presence-Time: 42 h; Self-Study: 93 h
- About [MAT-40-24-K-4]: The study achievement [U-Schein] proof of successful participation in the exercise classes (ungraded) must be obtained.
Examination achievement PL1
- Form of examination: oral examination (20-30 Min.)
- Examination Frequency: irregular (by arrangement)
- Examination number: 84141 ("p-adic Numbers")
Evaluation of grades
The grade of the module examination is also the module grade.
Contents
Competencies / intended learning achievements
Upon completion of this module, the students have studied extensions of a number system through p-adic numbers which are fundamental for number theory. In particular, they have studied their main properties and some simple applications. They are able to name and to prove the essential statements of the lecture as well as to classify and to explain the connections.
By completing the given exercises, the students have developed a skilled, precise and independent handling of the terms, propositions and techniques taught in the lecture. In addition, they have learnt how to apply these techniques to new problems, analyze them and develop solution strategies independently or by team work.
Literature
- F. Lorenz: Einführung in die Algebra II,
- N. Koblitz: p-adic Numbers, p-adic Analysis, and Zeta-Functions,
- I.B. Fenseko, S.V. Vostokov: Local Fields and their Extensions.
Registration
Registration for the exercise classes via the online administration system URM (https://urm.mathematik.uni-kl.de)
Requirements for attendance (informal)
More advanced knowledge from the courses [MAT-12-22-K-3] and [MAT-12-21-K-3] is useful but not necessarily required.
Modules:
Courses
Requirements for attendance (formal)
None
References to Module / Module Number [MAT-40-24-M-4]
Course of Study | Section | Choice/Obligation |
---|---|---|
[MAT-88.105-SG] M.Sc. Mathematics | Pure Mathematics | [WP] Compulsory Elective |
[MAT-88.706-SG] M.Sc. Mathematics International | Pure Mathematics | [WP] Compulsory Elective |
Module-Pool | Name | |
[MAT-GM-MPOOL-5] | General Mathematics (Introductory Modules M.Sc.) |
Notice