Part One The Basic Objects of Algebra
Chapter I Groups
1. Monoids
2. Groups
3. Normal subgroups
4. Cyclic groups
5. Operations of a group on a set
6. Sylow subgroups
7. Direct sums and free abelian groups
8. Finitely generated abelian groups
9. The dual group
10. Inverse limit and completion
11. Categories and functors
12. Free groups
Chapter II Rings
1. Rings and homomorphisms
2. Commutative rings
3. Polynomials and group rings
4. Localization
5. Principal and factorial rings
Chapter III Modules
1. Basic definitions
2. The group of homomorphisms
3. Direct products and sums of modules
4. Free modules
5. Vector spaces
6. The dual space and dual module
7. Modules over principal rings
8. Euler-Poincare maps
9. The snake lemma
10. Direct and inverse limits
Chapter IV Polynomials
1. Basic properties for polynomials in one variable
2. Polynomials over a factorial ring
3. Criteria for irreducibility
4. Hilbert's theorem
5. Partial fractions
6. Symmetric polynomials
7. Mason-Stothers theorem and the abe conjecture
8. The resultant
9. Power series
Part Two Algebraic Equations
Chapter V Algebraic Extensions
1. Finite and algebraic extensions
2. Algebraic closure
3. Splitting fields and normal extensions
4. Separable extensions
5. Finite fields
6. Inseparable extensions
Chapter VI Galois Theory
1. Galois extensions
2. Examples and applications
3. Roots of unity
4. Linear independence of characters
5. The norm and trace
6. Cyclic extensions
7. Solvable and radical extensions
8. Abelian Kummer theory
9. The equation X\