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Course, academic year 2018/2019
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Quadratic forms and class fields II - NMAG456
Title in English: Kvadratické formy a třídová tělesa II
Guaranteed by: Department of Algebra (32-KA)
Faculty: Faculty of Mathematics and Physics
Actual: from 2018 to 2018
Semester: summer
E-Credits: 3
Hours per week, examination: summer s.:2/0 Ex [hours/week]
Capacity: unlimited
Min. number of students: unlimited
State of the course: taught
Language: English, Czech
Teaching methods: full-time
Additional information: https://sites.google.com/site/vitakala/teaching/18kf
Guarantor: Mgr. Vítězslav Kala, Ph.D.
Class: M Mgr. MSTR
M Mgr. MSTR > Povinně volitelné
Classification: Mathematics > Algebra
Annotation -
Last update: doc. Mgr. et Mgr. Jan Žemlička, Ph.D. (09.05.2018)
Class field theory, which provides a substantial generalization of quadratic reciprocity law, forms the foundation for many advanced areas of number theory including Langlands program. In principle, it is concerned with the description of abelian extensions of number fields and p-adic numbers. In the course we will explain the main statements of this theory for global or local fields including the main tools for their proofs and various applications, mostly on the structure of number fields and quadratic forms. Specific choice of topics will depend on the interests of participants.
Course completion requirements - Czech
Last update: doc. Mgr. et Mgr. Jan Žemlička, Ph.D. (11.06.2019)

Předmět je zakončen ústní zkouškou.

Literature -
Last update: doc. Mgr. et Mgr. Jan Žemlička, Ph.D. (09.05.2018)

James A. Milne, Class Field Theory, online.

David A. Cox, Primes of the Form x^2+ny^2: Fermat, Class Field Theory, and Complex Multiplication, Wiley, 1989.

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

Requirements to the exam - Czech
Last update: Mgr. Vítězslav Kala, Ph.D. (21.09.2018)

Zkouška bude ústní s 30-60 minutami na přípravu jedné nebo dvou otázek, odpovídajících probrané látce na přednáškách.

Syllabus -
Last update: doc. Mgr. et Mgr. Jan Žemlička, Ph.D. (09.05.2018)

Completions, adeles and ideles

Main theorems of class field theory, Hilbert class field

Artin map

Proof tools: L-functions and group cohomology

Tchebotareff density theorem

Hasse local-global principle for quadratic forms

Higher reciprocity laws, Hilbert symbol

Local class fields: Lubin-Tate theory

 
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