|Galois theory and universal algebras|
|Teaching Staff in Charge|
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.
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.
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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.
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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.
Homework. Tests (33% x final grade). Oral exam (67% x final grade).