Mathematics BSc
Course description
2013. Algebra 3
Hours
lect+pc
Credits
lect+pc
Assessment Specialization Course code
lect/pc
Semester Status
2 + 2 2 + 3 exam +
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.