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

SYLLABUS

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

Semester	Hours: C+S+L	Category	Status	
3	2+2+0	specialty	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
This course aims to deepen the knowledge on Ring Theory and  Number Theory from a higher point of view. Introduction to Algebraic Number Theory. Developing problem solving skills (congruences, Diophantine equations).

IV. Course contents
Divisibility in integral domains. Prime numbers. congruences. Systems of linear congruences.
The group of units of Z_n. n-th power residues. The Legendre symbol. Quadratic reciprocity.
Quadratic fields and quadratic integers. Pitagorean numbers. About Fermat's Last Theorem.
Pell's equation. Noetherian rings and modules. Rings and modules of fractions. 
Algebraic number fields. Rings of algebraic integers. Valuation rings. Dedekind rings.
Cyclotomic fields. The Kronecker-Weber theorem. 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  (library)
2.	T.Albu, Ion D. Ion, Capitole de teoria algebrica a numerelor, Editura Academiei,  Bucuresti, 1984  (library)	

VI. Thematic of didactic activities per weeks
Schedule of the courses and seminars 
Courses:
1) Divisibility in integral domains. [1, Chap. 1, 2]
       1.1. Divisibility in Z. 
       1.2. Prime numbers
       1.3. Arithmetic functions
       1.4. Factorial rings
2) Congruences  [1, Chap. 3, 4]
       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  [1, chap. 5]
       3.1. The Legendre symbol
       3.2. Quadratic reciprocity
       3.3. The Jacobi symbol
4) Quadratic fields and rings of quadratic integers  [1, Sec. 13.1] [2, sec 3.4]
       4.1. Ramification
       4.2. Euclidean rings
5) Diophantine equations (1)   [1, sec. 17.1, 17.6]
       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                      [1, sec 17.2]
       6.3. Case n=3                     [1, sec. 17.8]
7) Diophantine equations (3)    [1, 17.5]
       7.1. Pell's equation 
8) Commutative rings    [2, Cap. 0]
       8.1. Noetherian rings and modules
       8.2. Rings and modules of fractions
9) Fields of algebraic numbers     [1, sec 12.1, 13.2]
       9.1. Algebraic extensions of fields
       9.2. Cyclotomic fields
10) Rings of algebraic integers (1)   [1, sec 12.2, 12.3] [2, cap 2]
      10.1. Algebraic integers
      10.2. Trace, norm, discriminant
11) Rings of algebraic integers (2)   [1, sec 12.3]
      11.1. Unique factorization
      11.2  Ramification ans degree      
12) Valuation rings    [2, sec. 2.4]
      12.1. Valued fields
      12.2. Discrete valuation rings
13) Dedekind rings        [2, 2.1-2.3]
      13.1. Definition
      13.2. Characterizations
14) Rings of algebraic integers (3)        
      14.1. Cyclotomic fields: intregi si ramificare   [1, sec 3.5]
      14.2. The Kronecker-Weber  theorem              [1, sec 3.6]
      14.3. The group of units of a ring of algebraic integers  [1, sec. 4.2]
Seminars:
1) Divisibility and primes
2) Congruences  and Sistems of linear congruences. 
3) Quadratic residues  and the Legendre and Jacobi symbols
4) Euclidean rings of quadratic integers         
5) Diophantine equations of degree 1 and 2
6)  Cases of  Fermat's Last Theorem
7)  Pell's equation 
8) Commutative Noetherian  rings    
9) Examples of algebraic extensions of fields
10) Algebraic integers. Calculation of trace, norm, discriminant
11) Factorization of algebraic integers 
12) Examples of valuation rings    
13) Examples of Dedekind rings        
14)  Calculation of the group of units of a ring of algebraic integers 

VII. Didactic methods used
Lectures, presentations, conversations, projects, exercises, individual study, homework assignments.	

VIII. Assessment
The activity ends with a written final. The exam subjects have theoretical questions from all the studied topics, and one problem, among the problems studied at the course and the seminar. There is an evaluation of the overall seminar activity, and the documentations prepared for the programming projects will be evaluated. The final grade is the mean of the grades mentioned above. The final grade = Homeworks 50%. Written Exam 50%.	

IX. Additional references
1.	Lang S., Algebra, Springer Verlag Berlin, 2002
2.	ROTMAN, J.: Advanced modern algebra, Prentice Hall, NJ 2002. 	

DECAN,                                                                                                  SEF DE CATEDRA,
Prof.dr. Leon Tâmbulea                                                                        Conf. dr. Simion Breaz

Date,							                              Course responsible,
2009,  September 24							      Prof. MARCUS Andrei, Ph.D.