Babes-Bolyai University of Cluj-Napoca
Faculty of Mathematics and Computer Science
Study Cycle: Master
SUBJECT
| MML1008 | Topics in Algebra III (for teachers education) |
| Teaching Staff in Charge |
Assoc.Prof. BREAZ Simion Sorin, Ph.D., bodo math.ubbcluj.ro |
| Aims |
|
Deepening the knowledge on Elementary Number Theory. Developing problem solving skills. Presenting some famous problems in Number Theory. Aproaching some classical results from a modern point of view. Presenting some arithmetic functions. Aproaching some number theory problems by using tools from modern algebra. Presenting some of the historical background of the subject evolution. |
| Content |
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1 Preliminaries
1.1 Mathematical Induction 1.2 The Binomial Theorem 2 Divisibility Theory in the Integers 2.1 The Division Algorithm 2.2 The Greatest Common Divisor 2.3 The Euclidean Algorithm 3 Primes and Their Distribution 3.1 The Fundamental Theorem of Arithmetic 3.2 The Sieve of Eratosthenes 3.3 The Goldbach Conjecture 4 The Theory of Congruences 4.1 Basic Properties of Congruence 4.2 Linear Congruences and the Chinese Remainder Theorem 5 Fermat@s Theorem 5.1 Fermat@s Little Theorem and Pseudoprimes 5.2 Wilson@s Theorem 6 Number-Theoretic Functions 6.1 The Sum and Number of Divisors 6.2 The Mobius Inversion Formula 6.3 The convolution product 7 Euler@s Generalization of Fermat@s Theorem 7.2 Euler@s Phi-Function 7.3 Euler@s Theorem and applications 8 Primitive Roots and Indices 147 8.1 The group of units of Z_n 8.2 Binomial congruences 9 Quadratic residues 9.1. The Legendre symbol 9.2. Quadratic reciprocity 9.3. The Jacobi symbol 10 Numbers of Special Form 10.1 Perfect Numbers 10.2 Mersenne Primes 10.3 Fermat Numbers 10.4 Fibonacci Numbers 11 Diophantine equations (1) 11.1. Equations of degree 1 11.2. Pithagorean numbers. Gauss integers 12 Representation of Integers as Sums of Squares 12.1 Sums of Two Squares 12.2 Sums of More Than Two Squares 13 Diophantine equations (2) 13.1. About Fermat@s Last Theorem 13.2. Case n=4 13.3. Case n=3. Euler integers 14. Diophantine equations (3) 14.1. Finite and Infinite Continued Fractions 14.2. Pell@s equation |
| References |
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1. K. Ireland, M. Rosen, A Classical Introduction to Modern Number Theory, Springer Verlag Berlin 1990
2. T. Albu, I. D. Ion, Capitole de teoria algebrica a numerelor, Editura Academiei, Bucuresti, 1984 3. S. Lang, Algebra, Springer Verlag Berlin, 2002 4. J. Rotman, Advanced modern algebra, Prentice Hall, NJ 2002 5. A. Marcus, Algebra [http://math.ubbcluj.ro/~marcus] 6. D. Burton, Elementary number theory, 6ed., MGH, 2007 |
| Assessment |
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A written final exam (grade E) and two tests at the seminar (grade T). The exam subjects have theoretical questions from all the studied topics, and one problem, among the
problems studied at the course and last 4 seminars. The tests subjects have practical questions (exercices and problems) from topics studied in first 10 weeks. The final grade is the weighted mean of the three grades mentioned above, conditioned by all the grades being at least 5 from 10. Otherwise, the exam will not be passed. The final grade = 60%E + 20%L + 20%R. |
| Links: | Syllabus for all subjects Romanian version for this subject Rtf format for this subject |