Babes-Bolyai University of Cluj-Napoca
Faculty of Mathematics and Computer Science
Study Cycle: Graduate
SUBJECT
| MML0006 | Galois Theory and Universal Algebras |
| Teaching Staff in Charge |
Assoc.Prof. PELEA Cosmin Razvan, Ph.D., cpelea math.ubbcluj.ro |
| Aims |
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To know some basics on field extensions and finite fields. An introduction to Galois theory. The study of some basic notions and results on universal algebras, applications to the algebraic structures studied in the previous semesters. |
| Content |
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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.
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. |
| References |
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1. S. BURRIS, H.P. SANKAPPANAVAR: A course in universal algebra, Springer-Verlag, 1981.
2. G. GRATZER: Universal algebra, Second Edition, Springer-Verlag, 1979. 3. I.D. ION, N. RADU: Algebra. Ed 4. Ed.Didactica si Pedagogica, 1990. 4. I.D. ION, N. RADU, C. NITA, D. POPESCU: Culegere de probleme de algebra, Ed. Didactica si Pedagogica, 1981. 5. S. LANG: Algebra, Addison-Wesley, Reading 1965. 6. C. NASTASESCU, C. NITA: Teoria calitativa a ecuatiilor algebrice, Ed. Tehnica, Bucuresti, 1979. 7. I. PURDEA: Tratat de algebra moderna vol. 2, Ed. Acad., 1982. 8. I. PURDEA, C. PELEA: Probleme de algebra, Ed. EIKON, 2008. 9. I. PURDEA, G. PIC: Tratat de algebra moderna, Vol.I, Ed. Acad., 1977. 10. I. PURDEA, I. POP, Algebra, Editura GIL, Zalau, 2003. |
| Assessment |
|
Homework. Tests (33% x final grade). Oral exam (67% x final grade). |
| Links: | Syllabus for all subjects Romanian version for this subject Rtf format for this subject |