Babes-Bolyai University of Cluj-Napoca
Faculty of Mathematics and Computer Science
Study Cycle: Master
SUBJECT

Code	Subject	
MML1008	Topics in Algebra III (for teachers education)	

Section	Sem.	Hours: C+S+L	Category	Type	
Didactic Mathematics	4	2+1+0	speciality	compulsory	

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	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 
	
Bibliography	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	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.