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Lecturer(s)
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Halaš Radomír, prof. Mgr. Dr.
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Chajda Ivan, prof. RNDr. DrSc.
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Kühr Jan, prof. RNDr. Ph.D.
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Course content
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1.Rings of residue classes and their invertible elements. Congruences of integers and their properties. 2.Primes and their properties, n-th prime, pi function, prime density. The law of asymptotic distribution of primes. 3.Congruence equations, linear congruence equations, continued fractions of rationals, systems of linear equations, linear diophantine equations. 4.Congruence equations of the second order, the symbol of Legendre, the lemma of Gauss, the reciprocity law. 5.Congruence equations in the prime power module, general congrunce equations. 6.Multiplicative groups of rings of residue classes , primitive roots. 7.Indices of elements and their properties, exponential and binomial congruence equations. 8.Continued fractions of irationals, their approximations by rationals. 9. The Hurwitz-Borel theorem, continued fractions of quadratic irationals, Pell's equations. 10.Algebraic and transcendental numbers, the Liouville theorem and constructions of transcendental numbers. 11.Numbers expressed as a sum of squares, the theorem of Lagrange on the sum of four squares. 12.The method of Schnirelmann on the sum of sequences, the hypothesis of Goldbach, the problem of Waring. 13.Minimal polynomial of an algebraic number and its construction. 14.Quadratic fields and their integers. 1. Constructions by ruler and linear. 2. Unsolvability of antic tasks. 3. Number fields, simple and finite algebraic extensions. Algebraicaly closed fields. 4. Galois group of algebraic extension. Cyclic and radical extension. 5. Solvability of algebraic equaitons in radicals. 6. Complex and hypercomplex numbers, quaternions, octets.
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Learning activities and teaching methods
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Lecture, Dialogic Lecture (Discussion, Dialog, Brainstorming)
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Learning outcomes
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Understand basics of classical number theory with applications in solving problems at secondary schools.
Learns important problems from number theory.
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Prerequisites
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unspecified
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Assessment methods and criteria
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Oral exam, Written exam
Credit: activity during seminars. Exam: the student has to understand the subject and be able to prove the principal results.
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Recommended literature
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B.L. van der Vaerden. (1991). Algebra I. Springer Verlag, 7-th ed.
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B.L. van der Vaerden. (1991). Algebra II. Springer Verlag, 7-th ed.
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Halaš, R. (1997). Teorie čísel. VUP Olomouc.
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Halaš, R. (2014). Úvod do teorie čísel. UP v Olomouci.
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Chajda I. (2000). Vybrané kapitoly z algebry. PřF UP Olomouc.
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Ireland M. (1987). Klasický úvod do moderní teorie čísel. Mir Moskva.
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Nathanson, M. B. (2000). Elementary methods in number theory. Springer.
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