Abstract Algebra - A Short Course

The book entitled ¿Abstract Algebra ¿ A Short Course¿ contains contains Permutation Group. Group of Symmetry. Dihedral group. Commutator group. Isomorphism Theorems. Automorphism. Characteristic subgroup. Conjugacy and G-Sets. Normal Series. Solvable groups. Nilpotent groups. Cyclic decomposition of permutation group. Alternating groups. Simplicity of An. Direct product, semi-direct product of groups. Sylows theorems. Groups of order and pq. Chapter 4 includes Ideals and Homomorphisms. Sum and direct sum of ideals. Maximal and prime ideals. Nilpotent and Nil ideals. Modules. Sub modules. Direct sums. R-homomorphisms and quotient modules. Completely reducible modules. Free modules. Unique factorization domain, principal ideal domains, Euclidean domains polynomial rings over unique factorization domains. Irreducible polynomials and Eisenstein criterion, Adjunction of roots, Algebraic extensions, Algebraically closed fields, Splitting fields, Normal extensions, multiple roots. Finite fields, separable extensions, Automorphism groups and fixed fields, Fundamental theorem of Galois theory, Fundamental theorem of Algebra. Roots of unity, cyclotomic polynomials, cyclic extension.