ABSTRACT ALGEBRA
ABSTRACT ALGEBRA
Syllabus - A. Bucki
- Mathematical Logic and Set Theory;
Equivalence Relations; Quotient Sets;
Partitions of Sets;
Functions; (Binary) Operations on Sets; Compatibility
of Operations with Equivalence Relations.
- Basic Properties of Integers - The Division
Algorithm, Euclid's Algorithm, Unique
Factorization in {\bf Z}; Functions
on a Finite Set - Operations on Permutations, $s$-Cycle,
Transposition, Standard Form;
Complex Numbers.
- Algebraic Structures: Groups - Examples,
Subgroups, Cyclic Groups, Normal Subgroups,
Quotient Groups, Solvable Groups, Group Homomorphisms, Isomorphisms,
Automorphisms, Kernel and Image of Homomorphism,
Cayley's Theorem; Rings - Commutative Rings,
Rings with Unity, Group of Invertibles, Subrings,
Ideals, Ring Homomorphisms; Integral Domains;
Fields, Extension Fields; Vector Spaces and Algebras.
- Applications of Algebraic Structures to:
Number Theory, The Euler and Fermat Theorems;
Galois Theory, Euclidean Constructions;
Coding Theory.