## Babes-Bolyai University of Cluj-Napoca Faculty of Mathematics and Computer Science Study Cycle: Graduate SUBJECT

 Code Subject MML0005 Algebra 3
 Section Semester Hours: C+S+L Category Type Mathematics 3 2+2+0 speciality compulsory Applied Mathematics 3 2+2+0 speciality compulsory
 Teaching Staff in Charge
 Prof. MARCUS Andrei, Ph.D.,  marcusmath.ubbcluj.roProf. PURDEA Ioan, Ph.D.,  purdeamath.ubbcluj.roLect. PELEA Cosmin Razvan, Ph.D.,  cpeleamath.ubbcluj.ro
 Aims This course deepens the study of rings and fields, by examining in detail their structure (lattice of subrings and ideals, generated ideal, characteristic, prime subfield). New examples and constructions will be presented (factor ring, products of rings, rings of fractions). Rings of polynomials in several variables are studied, with applications to algebraic equations. Arithmetic is studied in the general context of integral domains.
 Content Chapter I. THE STRUCTURE OF RINGS AND FIELDS 1. Rings and fields, subrings, homomorphisms. 2. The lattice of subrings and of ideals. Simple rings 3. Rings and ideals generated by a set. 4. Quotient rings. 5. The ring of residues modulo n. 6. Isomorphisms theorems. 7. Direct products of rings. 8. Prime subfields. Characteristic. 9. Rings of fractions. Chapter II. POLYNOMIALS AND ALGEBRAIC EQUATIONS. 1. Polynomials in one indeterminate. The universal property. 2. The formal derivative. Multiple roots. 3. Polynomials in n indeterminates. 4. Symmetric polynomials. The fundemental theorem. The Newton-Waring formulas. 5. Discriminant and resultant. 6. Algebraic equations. Chapter III. ARITHMETIC IN INTEGRAL DOMAINS 1. Divisibility. Prime and irreducible elements. 2. Factorial rings. 3. Principal ideal domains. The Chinese remainder theorem. 4. Euclidean domains. 5. Arithmetic in polynomial rings.
 References 1. I. PURDEA, I. POP, Algebra, Editura GIL, Zalau, 2003. 2. I.D. ION, N. RADU, Algebra (ed.4), Ed. Didactica si Pedagogica, Bucuresti 1991. 3. J. ROTMAN, Advanced modern algebra, Prentice Hall, NJ 2002. 4. G. CALUGAREANU, P. HAMBURG: Exercises in basic ring theory, Kluwer, Dordrecht 1998. 5. A. MARCUS, Algebra [http://math.ubbcluj.ro/~marcus] 6. J. SZENDREI, Algebra es szamelmelet, Tankonyvkiado, Budapest 1993. 7. M. BALINT, G. CZEDLI, A. SZENDREI: Absztrakt algebrai feladatok, Tankonyvkiado, Budapest1988. 8. G. SCHEJA, U. STORCH: Lehrbuch der Algebra 1,2, B.G. Teubner, Stuttgart 1994 9. M. ARTIN, Algebra, Birkhauser, Basel 1998. 10. I. PURDEA, C. PELEA, Probleme de algebra, EFES Cluj-Napoca 2005.
 Assessment Homework. Tests (25% x final grade). Oral exam (75% x final grade).
 Links: Syllabus for all subjects Romanian version for this subject Rtf format for this subject