Universitatea Babeş-Bolyai Cluj-Napoca
Facultatea de Matematică şi Informatică
Ciclul de studii: Masterat
Domeniul: Matematica
Programul de studii: Modelare Interdisciplinară - în limba maghiară
Limba de predare: Engleză

SYLLABUS

I. General data
Code	Subject	
MML1004	Commutative Rings and Number Theory	

Semester	Hours: C+S+L	Category	Status	
3	2+2+0	speciality	optional	

II. Full status faculty members
Name and	Scientific	Didactic	Chair	Type of activity	
surname	title	title		C	S	L	
MARCUS Andrei	Ph.D.	Prof.	Algebra	*			

Associated faculty members
Name and	Scientific	Institution	Type of	Type of activity	
surname	title		position	C	S	L	

III. Course objectives
Deepening the knowledge on Number Theory from a higher point of view.
Introduction to Algebraic Number Theory. Developing problem solving skills.

IV. Course contents
1) Divisibility in integral domains. 
       1.1. Divisibility in Z. 
       1.2. Prime numbers
       1.3. Arithmetic functions
       1.4. Factorial rings
2) Congruences 
       2.1. Sistems of linear congruences. The Chinese Remainder Theorem
       2.2. The group of units of Z_n
       2.3. n-th power residues
3) Quadratic residues
       3.1. The Legendre symbol
       3.2. Quadratic reciprocity
       3.3. The Jacobi symbol
4) Quadratic fields and rings of quadratic integers 
       4.1. Ramification
       4.2. Euclidean rings
5) Diophantine equations (1)  
       5.1. Equations of degree 1
       5.2. Pithagorean numbers
6) Diophantine equations (2)  
       6.1. About Fermat@s Last Theorem
       6.2. Case n=4  
       6.3. Case n=3  
7) Diophantine equations (3)  
       7.1. Pell@s equation 
8) Commutative rings  
       8.1. Noetherian rings and modules
       8.2. Rings and modules of fractions
9) Fields of algebraic numbers  
       9.1. Algebraic extensions of fields
       9.2. Cyclotomic fields
10) Rings of algebraic integers (1)  
      10.1. Algebraic integers
      10.2. Trace, norm, discriminant
11) Rings of algebraic integers (2)  
      11.1. Unique factorization
      11.2  Ramification ans degree      
12) Valuation rings
      12.1. Valued fields
      12.2. Discrete valuation rings
13) Dedekind rings 
      13.1. Definition
      13.2. Characterizations
14) Rings of algebraic integers (3) 
      14.1. Cyclotomic fields: intregi si ramificare 
      14.2. The Kronecker-Weber  theorem
      14.3. The group of units of a ring of algebraic integers 

V. Bibliography
1. K. Ireland, M. Rosen, A Classical Introduction to Modern Number Theory, Springer Verlag Berlin 1990
2. T.Albu, Ion D. Ion, Capitole de teoria algebrica a numerelor, Editura Academiei,  Bucuresti, 1984
3. Lang S., Algebra, Springer Verlag Berlin, 2002
4. ROTMAN, J.: Advanced modern algebra, Prentice Hall, NJ 2002
5. A. MARCUS: Algebra [http://math.ubbcluj.ro/~marcus]

VI. Thematic of didactic activities per weeks

VII. Didactic methods used

VIII. Assessment
Homeworks (20%). Exam. (80%)	

IX. Additional references

	Dean,							Chair head,

	Date,							Course responsible,
								Prof. MARCUS Andrei, Ph.D.