Course in Abstract Algebra

1. Group theory

• Subgroups and their cosets.
• Lagrange's theorem.
• Quotient groups. Isomorphism theorems.
• Actions of groups on sets.
• Orbits and stabilizers.
• Classification of finite rotation groups.
• Groups of prime power order. Sylow's theorems. 
• Elementary properties of solvable and nilpotent groups.
• Structure of finitely generated abelian groups.
• Symmetric and al­ternating groups.
• General linear groups.
• Examples of simple groups.
• Linear representations of groups.
• Maschke's theorem.
• Characters of representations.
• Orthogonality relations.
• Decomposition of the regular representation.
• Conjugacy classes and class functions.
• Character tables for groups of small order.
• Duality of finite abelian groups.

2. Rings and Modules

• Factor rings and factor modules.
• Isomorphism theorems for rings and modules
• The Chinese remainder theorem.
• Direct sums and direct products.
• Free modules.
• Invariance of the basis numbers for commutative rings.
• The ascending and descending chain conditions.
• Modules with composition series.
• The Jordan-H\"older theorem.
• The Jacobson radical.
• Artinian and noetherian rings.
• Hilbert's basis theorem.
• Simple and semisimple rings.
• The Artin-Wedderburn structure theory.
• Finite dimensional division algebras over the reals.

3. Commutative ring theory

• Prime and maximal ideals in commutative rings.
• A characterization of the nilradical in terms of prime ideals.
• Local rings.
• Rings of fractions.
• Prime ideals in the rings of fractions.
• Integral dependence.
• The going-up theorem.
• The classical Krull dimension.
• Finitely generated algebras over a field.
• Hilbert's Nullstellensatz.
• Discrete valuations and discrete valuation rings.
• Fractional ideals.
• Dedekind domains.
• Rings of algebraic integers.
• Unique factorization in the principal ideal domains.
• Finitely generated modules over principal ideal domains.

4. Fields

• Algebraic and finite field extensions.
• Extensions of isomorphisms.
• Splitting fields of polynomials.
• Algebraic closure.
• Finite fields.
• The multiplicative group of a finite field.
• Normal extensions.
• Galois groups.
• A theorem about the fixed field of a finite automorphism group.
• The main result of the Galois theory.
• The primitive element theorem.
• Cyclotomic extensions.
• Radical extensions.
• Solvability of algebraic equations in one indeterminate.