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

Code	Subject	
MML0016	Rings and Fields	

Section	Sem.	Hours: C+S+L	Category	Type	
Mathematics - in Romanian	4	2+1+0	speciality	optional	
Mathematics-Computer Science - in Romanian	4	2+1+0	speciality	optional	

Teaching Staff in Charge	Prof. MARCUS Andrei, Ph.D., marcus@math.ubbcluj.ro
Assoc.Prof. PELEA Cosmin Razvan, Ph.D., cpelea@math.ubbcluj.ro	
Aims	Study of polynomial rings, divisibility in integral domains and an introduction to field theory.	
Content	1. Semigroups and rings of fractions.
2. Polynomial rings. Construction of polynomial rings in one indeterminate, 
universal property, algebraic and transcendental elements. Division algorithm for polynomials. Polynomial functions. Polynomials in several indeterminates. The field of rational fractions. Symmetric polynomials, the fundamental theorem of symmetric polynomials, symmetric rational fractions.
3. Divisibility in commutative monoids and in integral domains. Divisibility relation and 
associated divisibility relation. Greatest common divisor and least common multiple. 
Prime elements and irreducible elements. Factorial semigroups. Euclidean domains. Principal ideal domains. Divisibility in polynomial rings. Prime ideals and maximal ideals. 
4. Fields. Characteristic of a ring. Determination of prime fields. Field extensions, 
degree of an extension. Algebraic extensions, adjunction. Minimal polynomial of 
an algebraic element. Adjunction of an algebraic element. The field of algebraic elements. Adjunction of a root. Splitting field of a polynomial. The fundamental theorem of Algebra.	
Bibliography	1. I. PURDEA, G. PIC: Tratat de algebra moderna, Vol.I, Ed. Acad.,1977. 
2. I. PURDEA: Tratat de algebra moderna vol. 2, Ed.Acad., 1982. 
3. I. PURDEA: Culegere de probleme de teoria grupurilor, Univ. din Cluj, 1985. 
4. I. PURDEA, I. POP:  Algebra, Ed. GIL, Zalau, 2003. 
5. I.D. ION, N. RADU: Algebra, ed. 4, Ed.Didactica si Pedagogica, 1990. 
6. G. CALUGAREANU: Culegere de probleme de inele, Univ. din Cluj, 1978. 
7. I.D. ION, N. RADU, C. NITA, D. POPESCU: Culegere de probleme de algebra, Ed. Didactica si Pedagogica, 1981.
8. N. JACOBSON:  Basic algebra vol. I, II, Freeman, San-Francisco  1984.
9. E. FRIED: Altalanos algebra, Tankonyvkiado, Budapest  1981.
10. A. MARCUS : Algebra [http://math.ubbcluj.ro/~marcus]
11. L. FUCHS: Algebra, Tankonyvkiado, Budapest   1992.
12. G. SCHEJA, U. STORCH: Lehrbuch der Algebra 1,2, B.G. Teubner, Stuttgart  1994.
13. M. ARTIN:   Algebra, Birkhauser, Basel  1998.

Assessment	Homework. Tests (25% x final grade). Oral exam (75% x final grade).