| Galois theory and universal algebras |
ter |
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| 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 |
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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 |
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Galois Theory. Separable extensions and normal extensions. Galois group. Finite fields.
Wedderburn's theorem. Determination of finite fields and of subfields of a finite field. Solvable groups. Characterization of equations solvable by radicals. The fundamental theorem of Galois Theory. Algebraically closed fields. 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. PURDEA I., PIC GH., Tratat de algebra moderna, Vol.I, Ed. Acad.,1978.
2. PURDEA I.,Tratat de algebra moderna, vol.II, Ed.Acad.,1982. 3. ION I.D., RADU N., Algebra, Ed. Did. si Ped., 1990. 4. ANDRICA, D., DUCA, D., PURDEA, I., Matematica de baza, Ed. Studium, Cluj-Napoca, 2001. |
| Assessment |
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Oral exam. |