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Number theory1L (mm1c1se1)
Syllabus designed by:
Classical and linear algebra, elementary number theory.
To present the basic notions and methods of abstract algebra required for applications.
- P. M. Cohn: Classic algebra. Wiley, 2000
- 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.