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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)
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Předmět je zakončen ústní zkouškou. Last update: Žemlička Jan, doc. Mgr. et Mgr., Ph.D. (10.06.2019)
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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)
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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)
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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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