MML1004 | Commutative Rings and Number Theory |
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 |