Subjects

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Basic algebraic number theory - NMAG472

Title:	Základní algebraická teorie čísel
Guaranteed by:	Department of Algebra (32-KA)
Faculty:	Faculty of Mathematics and Physics
Actual:	from 2024
Semester:	summer
E-Credits:	3
Hours per week, examination:	summer s.:2/0, Ex [HT]
Capacity:	unlimited
Min. number of students:	unlimited
4EU+:	no
Virtual mobility / capacity:	no
State of the course:	taught
Language:	English, Czech
Teaching methods:	full-time
Additional information:	https://www1.karlin.mff.cuni.cz/~kala/web/teaching/24batc

Guarantor:	doc. Mgr. Vítězslav Kala, Ph.D.
Teacher(s):	doc. Mgr. Vítězslav Kala, Ph.D.
Class:	M Mgr. MMIB > Povinně volitelné M Mgr. MSTR > Povinně volitelné
Classification:	Mathematics > Algebra
Incompatibility :	NMAG430
Is incompatible with:	NMAG430

Opinion survey results Examination dates SS schedule Noticeboard

Annotation -

Algebraic number theory studies the structure of number fields and forms the basis for most of advanced areas of number theory. In the course we will develop its main tools that are connected to algebraic integers, prime ideals, ideal class group, and unit group, and we will richly illustrate them by examples.

Last update: Žemlička Jan, doc. Mgr. et Mgr., Ph.D. (17.05.2024)

Literature -

Daniel A. Marcus, Number Fields, Universitext, 2018.

James A. Milne, Algebraic Number Theory, online.

Serge Lang, Algebraic Number Theory, GTM 110, 1994.

H. Cohen, A course in computational algebraic number theory, Springer-Verlag, Berlin 1996.

A. Frőhlich, M. J. Taylor, Algebraic number theory, Cambridge University Press, Cambridge 1991.

Last update: Žemlička Jan, doc. Mgr. et Mgr., Ph.D. (17.05.2024)

Requirements to the exam

There will be an oral exam with questions corresponding to the material covered in the course. See course website for more information.

Last update: Kala Vítězslav, doc. Mgr., Ph.D. (26.02.2025)

Syllabus -

Review of basic notions

Algebraic integers

Norm, trace, discriminant

Prime factorization, ramification and splitting

Geometry of numbers, Minkowski bound

Finiteness of class group

Dirichlet unit theorem

Last update: Žemlička Jan, doc. Mgr. et Mgr., Ph.D. (17.05.2024)