Course for international guest/part time students

Faculty

Faculty of Science

Organization

TTK Department of Algebra and Number Theory

Code

algsza1u0um17gm

Title

Algebraic number theory (p)

Usual semester

Autumn

Published semester

2026/27/1

ECTS

3

Language

en

Learning outcomes

·      Knowledge: Knowledge of basic concepts, results and methods in the field. ·      Ability: Application of knowledge in the field, understanding of interrelationships and problem solving. ·      Attitude: Desire to improve mathematical knowledge and to learn as much as possible, and to apply knowledge as widely as possible. Autonomy and responsibility: Formulate and analyse mathematical questions independently and evaluate the limits of their applicability responsibly.

Course content

Quadratic reciprocity. Integral elements in ring extensions, integral closure, trace, norm, discriminant, integral basis. Dedekind domains, ideal theory, unique factorization. Class group, comparison with the Picard group of schemes, Minkowski's estimate of the class number, Dirichlet's theorem on units in the ring of integers. Hilbert ramification theory, decomposition- and ramification subgroups. Cyclotomic fields, Fermat's Last Theorem for regular primes. Localisation at prime ideals, discrete valuation rings, completion, p-adic numbers, inverse limit, direct limit. Ostrowski's theorem on completions of Q. Hensel's lemma, Teichmüller representatives. Local fields, p-adic log and exp, description of the multiplicative group. Ring of Witt vectors. Henselian fields, extension of valuations. Explicit description of unramified and tamely ramified extensions, ramification subgroups, Hasse-Herbrand function, connection with the norm. The field of norms (w/o proofs). Local and global Kronecker-Weber Theorem (w/o proofs) and their connection to class field theory. Necessary prior knowledge: basic group-, ring-, and Galois theory

Assessment method

term mark

Bibliography

·       J. Neukirch, Algebraische Zahlentheorie, Springer (1992). ·       J.-P. Serre, Local fields, Graduate Texts in Math. 67, 2nd Ed.  (1995). ·       S. Lang, Algebraic Number Theory, Graduate Texts in Math. 110, 2nd Ed. (1994). ·       W. J. Milne, Algebraic Number Theory, http://jmilne.org/math/CourseNotes/ant.html

Recommended bibliography

·       J. Neukirch, Algebraische Zahlentheorie, Springer (1992). ·       J.-P. Serre, Local fields, Graduate Texts in Math. 67, 2nd Ed.  (1995). ·       S. Lang, Algebraic Number Theory, Graduate Texts in Math. 110, 2nd Ed. (1994). W. J. Milne, Algebraic Number Theory, http://jmilne.org/math/CourseNotes/ant.html

Programmes of the course