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Lecturer(s)
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Halaš Radomír, prof. Mgr. Dr.
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Course content
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The aim of the course is to deepen knowledge of elementary number theory, to place it in a broader context, and to further introduce students to more advanced parts of classical number theory. Emphasis will be placed on connecting the material with other areas of mathematics. 1. Decomposition of numbers into sums of squares. Lagrange's theorem on the sum of four squares. 2. Quadratic fields and integer algebraic numbers. Integer algebraic numbers in quadratic fields. Divisibility in integer algebraic numbers. 3. Schnirelmann's method of adding sequences, Goldbach's hypothesis, Waring's problem. 4. Quadratic reciprocity law and its proofs. 5. Foundations of finite fields. 6. Multiplicative characters in finite fields and their importance in solving congruent equations. Jacobi and Gauss sums. 7. Cubic and biquadratic reciprocity laws. 8. Riemann hypothesis and zeta-function. 9. Algebraic number theory, uniqueness of decompositions. 10. Diophantine equations and methods of their solution.
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Learning activities and teaching methods
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Monologic Lecture(Interpretation, Training)
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Learning outcomes
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Understand more advanced parts of number theory and their applications in various areas of mathematics.
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Prerequisites
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unspecified
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Assessment methods and criteria
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Oral exam, Student performance, Written exam
Active participation in exercises, successful completion of the written exam.
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Recommended literature
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Halaš R. (1997). Number Theory. Olomouc.
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Ireland M. (1987). Classical introduction to modern number theory. Moscow.
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Ireland M. (1987). Klasický úvod do moderní teorie čísel. Moskva.
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Nathanson M.B. (2000). Elementary methods in number theory. Springer.
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Radomír Halaš. (1997). Teorie čísel. Olomouc.
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