Mathematics BSc
Course description
2013.
Course description
2013.
Algebra 3
Hours lect+pc |
Credits lect+pc |
Assessment | Specialization | Course code lect/pc |
Semester | Status |
---|---|---|---|---|---|---|
2 + 2 | 2 + 3 | exam + term grade |
applied math. | mm1c1al3a mm1c2al3a |
3 | compulsory |
Course coordinator
Strong | Weak | Prerequisites | |
---|---|---|---|
Practice class | |||
Strong:
Algebra2L
(mm1c1al2)
| |||
Strong:
Number theory1L
(mm1c1se1)
| |||
Lecture | |||
Weak:
practice class
|
Prerequisites
Classical and linear algebra, elementary number theory.
Course objectives
To present the basic notions and methods of
abstract algebra required for applications.
Literature
- P. M. Cohn: Classic algebra. Wiley, 2000
Syllabus
- Groups. Examples. Order of an element, cyclic groups. Subgroups, cosets, Lagrange's theorem.
- Permutation groups, orbits and stabilizers, transitivity, Cayley's theorem.
- Homomorphism, normal subgroups, quotient groups, isomorphism theorems.
- Direct product, the fundamental theorem of finite abelian groups.
- Rings. Subrings and ideals, simple rings, annihilator.
- Characteristic, prime field.
- Theorems of Wedderburn-Artin and Frobenius.
- Galois theory. Degree of a field extension, simple algebraic extensions, finite and algebraic extension, the field of algebraic numbers is algebraically closed.
- Splitting field, normal extensions, Galois group, the fundamental theorem of Galois theory.
- Finite fields, additive and multiplicative group, Galois group, subfields. Wedderburn's theorem.
- Introduction to error-correcting codes: Hamming distance, linear codes, perfect codes, BCH codes, decoding methods.