## "Babes-Bolyai" University of Cluj-Napoca Faculty of Mathematics and Computer Science

 Rings and fields
 Code Semes-ter Hours: C+S+L Credits Type Section MA003 3 2+2+0 6 compulsory Matematică MA003 3 2+2+0 6 compulsory Matematică-Informatică MA003 3 2+2+0 6 compulsory Matematici Aplicate
 Teaching Staff in Charge
 Prof. PURDEA Ioan, Ph.D., purdea@math.ubbcluj.roProf. MARCUS Andrei, Ph.D., marcus@math.ubbcluj.roLect. SACAREA Cristian, Ph.D., csacarea@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.
 References 1. PURDEA, I, PIC, GH., Tratat de algebra moderna, Vol.I, Ed. Acad.,1977. 2. PURDEA, I., Tratat de algebra moderna, Ed.Acad., 1982. 3. ION, I.D., RADU N., Ed.Did. si Ped., 1990. 4. PURDEA, I., Culegere de probleme de teoria grupurilor, Univ. din Cluj, 1985. 5. CALUGAREANU, GR., Culegere de probleme de inele, Univ. din Cluj, 1978. 6. ION, I.D., RADU, N., NITA, C.,POPESCU, D., Culegere de probleme de algebra, Ed. Did. si Ped., 1981. 7. ANDRICA, D., DUCA, D., PURDEA, I., Matematica de baza, Ed. Studium , Cluj-Napoca, 2001.
 Assessment Oral exam.