"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.ro
Prof. MARCUS Andrei, Ph.D., marcus@math.ubbcluj.ro
Lect. 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.