Universität Wien
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250069 VO Algebraic number theory (2012W)

6.00 ECTS (4.00 SWS), SPL 25 - Mathematik

Details

Language: German

Examination dates

Lecturers

Classes (iCal) - next class is marked with N

  • Thursday 04.10. 15:00 - 17:00 Seminarraum
  • Friday 05.10. 13:00 - 15:00 Seminarraum
  • Thursday 11.10. 15:00 - 17:00 Seminarraum
  • Friday 12.10. 13:00 - 15:00 Seminarraum
  • Thursday 18.10. 15:00 - 17:00 Seminarraum
  • Friday 19.10. 13:00 - 15:00 Seminarraum
  • Thursday 25.10. 15:00 - 17:00 Seminarraum
  • Thursday 08.11. 15:00 - 17:00 Seminarraum
  • Friday 09.11. 13:00 - 15:00 Seminarraum
  • Thursday 15.11. 15:00 - 17:00 Seminarraum
  • Friday 16.11. 13:00 - 15:00 Seminarraum
  • Thursday 22.11. 15:00 - 17:00 Seminarraum
  • Friday 23.11. 13:00 - 15:00 Seminarraum
  • Thursday 29.11. 15:00 - 17:00 Seminarraum
  • Friday 30.11. 13:00 - 15:00 Seminarraum
  • Thursday 06.12. 15:00 - 17:00 Seminarraum
  • Friday 07.12. 13:00 - 15:00 Seminarraum
  • Thursday 13.12. 15:00 - 17:00 Seminarraum
  • Friday 14.12. 13:00 - 15:00 Seminarraum
  • Thursday 10.01. 15:00 - 17:00 Seminarraum
  • Friday 11.01. 13:00 - 15:00 Seminarraum
  • Thursday 17.01. 15:00 - 17:00 Seminarraum
  • Friday 18.01. 13:00 - 15:00 Seminarraum
  • Thursday 24.01. 15:00 - 17:00 Seminarraum
  • Friday 25.01. 13:00 - 15:00 Seminarraum
  • Thursday 31.01. 15:00 - 17:00 Seminarraum

Information

Aims, contents and method of the course

Number theory is an area in mathematics dealing with properties
of integers in the broadest sense. Algebraic number theory studies
the arithmetic of algebraic number fields — the ring of integers
in the number field, the ideals and units in the ring of integers,
the extent to which unique factorization holds, and so on.
Among other things, the theory arose out of the study of
Diophantine equations.

The chapters are as follows:

Chapter 1: Integral ring extensions, in particular global fields and their
rings of interegs, norm, trace and discriminant.

Chapter 2: Ideals in Dedekind rings, fractional ideals, ideal class group,
unique factorization.

Chapter 3: Finiteness of the class number, Minkowski-Theory,
rings of integers as lattices, special case calss number $1$ fields.

Chapter 4: Dirichlet's Unit Theorem, the analytic class number formula.

Chapter 5: Decomposition and ramification, in general and for Galois
extensions. Ramification and discriminant.

Chapter 6: Cyclotomic fields and their rings of integers, units, and the
Fermat equation.

Chapter 7: Absolute values and local fields, completions, the adelic
viewpoint.

Assessment and permitted materials

Written exam or oral exam after the end of the lecture.

Minimum requirements and assessment criteria

Introduction to algebraic number theory, its motivations, methods, developments and results.

Examination topics

Reading list

[BUR] D. Burde, Commutative Algebra, 2009.
[COH] H. Cohen, A course in computational algebraic number theory, 1993.
[KOC] H. Koch, Algebraic number theory, 1997.
[LAN] S. Lang, Algebraic number theory, 1994.
[NEU] J. Neukirch, Algebraic number theory, 1999.
(WAS] L. C. Washington, Introduction to cyclotomic fields, 1997.

Association in the course directory

MALZ

Last modified: Mo 07.09.2020 15:40