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

SUBJECT

Code
Subject
MML1004 Commutative Rings and Number Theory
Section
Semester
Hours: C+S+L
Category
Type
Mathematics
3
2+1+0
speciality
optional
Didactic Mathematics - in Hungarian
3
2+1+0
speciality
optional
Interdiciplinary Computational - in Hungarian
3
2+1+0
speciality
optional
Teaching Staff in Charge
Prof. MARCUS Andrei, Ph.D.,  marcusmath.ubbcluj.ro
Aims
Deepening the knowledge on Number Theory from a higher point of view.
Introduction to Algebraic Number Theory. Developing problem solving skills.

Content
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
References
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]

Assessment
Homeworks (20%). Exam. (80%)
Links: Syllabus for all subjects
Romanian version for this subject
Rtf format for this subject