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

Code	Subject	
MML0006	Galois Theory and Universal Algebras	

Section	Semester	Hours: C+S+L	Category	Type	
Mathematics - in Romanian	6	2+1+0	speciality	optional	
Applied Mathematics	6	2+1+0	speciality	optional	

Teaching Staff in Charge	Lect. PELEA Cosmin Razvan, Ph.D., cpelea@math.ubbcluj.ro	
Aims	An introduction to Galois theory. The study of notions and basic results of the theory of universal algebras applied to the algebraic structures studied in the previous semesters, completed with new properties.	
Content	Galois Theory. Field extensions. Separable extensions and normal extensions. Algebraically closed fields.  Finite fields. Wedderburn's theorem. Determination of finite fields and of subfields of a finite field. Galois group. The fundamental theorem of Galois Theory. Solvable groups. Characterization of equations solvable by radicals. Constructibility by ruler and compass. 
Universal algebras. n-ary operations and universal algebras. Homomorphisms. Stable subsets, subalgebras. The lattice of subalgebras, generated subalgebra. Particular cases: generated subsemigroup, generated subgroup, generated subring, generated submodule. Algebraic closure systems and operators. Direct products of universal algebras. Homomorphic relations. Quotient algebraic congruences. The lattice of congruences. The connection between the congruences of a group and its normal subgroups. The connection between the congruences of a ring and its ideals. Factorization of a homomorphism through a surjective or injective homomorphism. The isomorphism theorems for universal algebras and deduction of the isomorphism theorems for groups and rings.	
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, I. POP, Algebra, Editura GIL, Zalau, 2003.
4. S. LANG: Algebra, Addison-Wesley, Reading  1965.
5. M.H. FENRICK: Introduction to the Galois Correspondence, 2nd Edition, Birkhauser, Basel 1996.
6. I.D. ION, N. RADU: Algebra. Ed 4. Ed.Didactica si Pedagogica, 1990. 
7. I.D. ION, N. RADU, C. NITA, D. POPESCU: Culegere de probleme de algebra, Ed. Didactica si Pedagogica, 1981.
8. M. BALINT, G. CZEDLI, A. SZENDREI: Absztrakt algebrai feladatok, Tankonyvkiado, Budapest  1988.
9. A. MARCUS : Algebra [http://math.ubbcluj.ro/~marcus]
10. L. FUCHS: Algebra, Tankonyvkiado, Budapest   1992.
11. G. SCHEJA, U. STORCH: Lehrbuch der Algebra 1,2, B.G. Teubner, Stuttgart  1994.
	
Assessment	Homework. Tests (33% x final grade). Oral exam (67% x final grade).