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Course, academic year 2018/2019
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Quadratic forms and class fields I - NMAG455
Title in English: Kvadratické formy a třídová tělesa I
Guaranteed by: Department of Algebra (32-KA)
Faculty: Faculty of Mathematics and Physics
Actual: from 2018 to 2018
Semester: winter
E-Credits: 3
Hours per week, examination: winter s.:2/0 Ex [hours/week]
Capacity: unlimited
Min. number of students: unlimited
State of the course: taught
Language: 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. (14.05.2019)
Quadratic forms with integral coefficients form a central part of number theory - for example, the study of primes represented by the form x^2+ny^2 gradually led to the development of many key tools in algebraic number theory, ranging from the study of number fields to the theory of class fields and modular forms. The goal of the course is to explain the basics of the arithmetic theory of quadratic forms, in particular with focus on the question of representability of integers including applications of class field theory.
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)

Leonard Eugene Dickson, Modern Elementary Theory of Numbers, Chicago, 1939.

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

Manjul Bhargava, On the Conway-Schneeberger fifteen theorem, Contemp. Math. 272, 27 - 37.

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)

Basic notions: equivalence of quadratic forms, determinant, associated matrix and lattice, reduction of forms

Ternary forms, 3- and 4-square theorems, universal diagonal forms, 15 theorem

Binary forms: composition and form class group, genus theory

Isomorphism of ideal and form class groups

Hilbert class field and primes of the form x^2+ny^2

 
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