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Course, academic year 2024/2025
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Quadratic forms and class fields I - NMAG455
Title: Kvadratické formy a třídová tělesa I
Guaranteed by: Department of Algebra (32-KA)
Faculty: Faculty of Mathematics and Physics
Actual: from 2024
Semester: winter
E-Credits: 3
Hours per week, examination: winter 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://sites.google.com/view/shman/quadratic-forms-and-class-fields-i-winter-2223
Guarantor: doc. Mgr. Vítězslav Kala, Ph.D.
Teacher(s): Pavlo Yatsyna, Ph.D.
Class: M Mgr. MSTR
M Mgr. MSTR > Povinně volitelné
Classification: Mathematics > Algebra
Annotation -
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. The course may not be taught every academic year.
Last update: Žemlička Jan, doc. Mgr. et Mgr., Ph.D. (14.05.2019)
Course completion requirements - Czech

Předmět je zakončen ústní zkouškou.

Last update: Žemlička Jan, doc. Mgr. et Mgr., Ph.D. (10.06.2019)
Literature -

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.

Last update: Žemlička Jan, doc. Mgr. et Mgr., Ph.D. (09.05.2018)
Requirements to the exam - Czech

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

Last update: Kala Vítězslav, doc. Mgr., Ph.D. (21.09.2018)
Syllabus -

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

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