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Algebraic Number Theory - NMAG430

Title:	Algebraická teorie čísel
Guaranteed by:	Department of Algebra (32-KA)
Faculty:	Faculty of Mathematics and Physics
Actual:	from 2023
Semester:	summer
E-Credits:	6
Hours per week, examination:	summer s.:3/1, C+Ex [HT]
Capacity:	unlimited
Min. number of students:	unlimited
4EU+:	no
Virtual mobility / capacity:	no
State of the course:	taught
Language:	Czech, English
Teaching methods:	full-time
Teaching methods:	full-time
Additional information:	https://sites.google.com/view/pyatsyna/teaching/algebraic-number-theory/202324

Guarantor:	doc. Mgr. Vítězslav Kala, Ph.D.
Class:	M Mgr. MMIB M Mgr. MMIB > Povinně volitelné M Mgr. MSTR M Mgr. MSTR > Povinně volitelné
Classification:	Mathematics > Algebra

Opinion survey results Examination dates SS schedule Noticeboard

Annotation -

Last update: doc. Mgr. et Mgr. Jan Žemlička, Ph.D. (07.12.2018)

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, unit group, and subgroups of the Galois group, including basics of p-adic numbers and applications to Diophantine equations.

Course completion requirements

Last update: doc. Mgr. et Mgr. Jan Žemlička, Ph.D. (16.02.2023)

The course requires an oral exam and credit for the exercises. The credit for the exercises "zapocet" will be awarded for successfully solving several sets of homework problems. Zapocet is not needed for taking the exam.

Literature -

Last update: doc. Mgr. et Mgr. Jan Žemlička, Ph.D. (07.12.2018)

James A. Milne, Algebraic Number Theory, online.

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

E.I. Borevič, I.R. Šafarevič: Number Theory, Academic Press 1966.

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.

Requirements to the exam

Last update: doc. Mgr. et Mgr. Jan Žemlička, Ph.D. (16.02.2023)

The exam is oral with approx. 60 minutes time for preparation for 1 or 2 questions corresponding to the material covered by the course.

Syllabus -

Last update: doc. Mgr. et Mgr. Jan Žemlička, Ph.D. (07.12.2018)

Algebraic integers

Dedekind domains

Prime factorization, ramification and splitting

Geometry of numbers, Minkowski bound

Finiteness of class group

Dirichlet unit theorem, regulator

Cyclotomic fields, Diophantine equations

p-adic numbers

Ramification and inertia group, Frobenius element