## 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