Module Handbook

  • Dynamischer Default-Fachbereich geändert auf MAT

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

Notice

Without a proof of successful participation in the exercise classes, only 3 credit points will be awarded for the module.

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

  • construction of p-adic numbers,
  • p-adic integers, units,
  • p-adic topology,
  • Hensel's lemma,
  • algebraic closure,
  • Newton polygon,
  • inertia and ramification groups

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.)