M8195 Number theory seminar

Faculty of Science
Autumn 2013
Extent and Intensity
0/2. 2 credit(s) (příf plus uk k 1 zk 2 plus 1 > 4). Type of Completion: z (credit).
Teacher(s)
Mgr. Michal Bulant, Ph.D. (lecturer)
prof. RNDr. Radan Kučera, DSc. (lecturer)
Guaranteed by
prof. RNDr. Radan Kučera, DSc.
Department of Mathematics and Statistics – Departments – Faculty of Science
Supplier department: Department of Mathematics and Statistics – Departments – Faculty of Science
Timetable of Seminar Groups
M8195/01: Thu 11:00–12:50 M5,01013, M. Bulant, R. Kučera
Prerequisites (in Czech)
Je vhodné absolvování předmětů Algebra II.
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
Course objectives
In this semester we shall start to study the book of D.A.Cox which is devoted to the following problem: for a given positive integer n determine which primes are of the form p = x^2 + n . y^2 for integers x, y. If n=1 then it is possible to show quite elementarily that these primes are exactly the primes not congruent to 3 modulo 4. If n=2 then these primes are exactly the primes not congruent to 5 or 7 modulo 8. A solution of this problem by elementary methods is given only for a few values of n. To answer this question for a general n is very difficult problem whose solution needs very deep results of number theory. The Cox's book nicely shows the motivation: why the mathematicians were led to construct and work with very abstract objects of algebraic number theory.
At the end of this course, students should be able to: explain the problems of expressing primes by a few easiest binary quadratic forms.
Syllabus
  • 1. Fermat, Euler and quadratic reciprocity;
  • 2. Lagrange, Legendre a quadratic forms;
  • 3. Gauss, composition anda genera;
  • 4. Cubic and biquadratic reciprocity.
Literature
  • COX, David A. Primes of the form x² + ny² : Fermat, class field theory, and complex multiplication. New York, N.Y.: John Wiley & Sons, 1989, xi, 351. ISBN 0471190799. info
Bookmarks
https://is.muni.cz/ln/tag/PříF:M8195!
Teaching methods
Lectures, homeworks.
Assessment methods
Credit will be given in case of active work in seminars: the study of the mentioned book during the term, regular delivering of solved homeworks.
Language of instruction
Czech
Further Comments
Study Materials
The course is taught each semester.
The course is also listed under the following terms Autumn 2007 - for the purpose of the accreditation, Spring 2008 - for the purpose of the accreditation, Spring 2011 - only for the accreditation, Autumn 2010 - only for the accreditation, Spring 2005, Autumn 2005, Spring 2006, Autumn 2006, Spring 2007, Autumn 2007, Spring 2008, Autumn 2008, Spring 2009, Autumn 2009, Spring 2010, Autumn 2010, Spring 2011, Autumn 2011, Spring 2012, Autumn 2011 - acreditation, spring 2012 - acreditation, Autumn 2012, Spring 2013, Spring 2014, Autumn 2014, Spring 2015, Autumn 2015, Spring 2016, Autumn 2016, Spring 2017, autumn 2017, spring 2018, Autumn 2018, Spring 2019, Autumn 2019, Spring 2020, Autumn 2020, Spring 2021, autumn 2021, Spring 2022, Autumn 2022, Spring 2023, Autumn 2023, Spring 2024, Autumn 2024, Spring 2025.
  • Enrolment Statistics (Autumn 2013, recent)
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