Contents.

Chapter 0. Preliminaries.

0.1 A short note on proofs

0.2 Sets and equivalence relations.

This chapter is assumed to be known. Only a short review will be given in the course.

Chapter 1. The integers.

1.1 Mathematical induction.

1.2 The division algorithm.

This chapter is assumed to be known. Only a short review will be given in the course.

Chapter 2. Groups.

2.1 The integers modulo n and symmetries.

2.2 Definitions and examples.

2.3 Subgroups.

Recommended exercises: 1, 3, 23, 26, 29, 30, 31, 38, 40, 41, 44, 45, 46, 53.

Chapter 3. Cyclic groups.

3.1 Cyclic subgroups.

Recommended exercises: 1, 2, 4, 10, 13, 14, 26, 27, 28, 30, 31, 34, 37.

Chapter 4. Permutaion groups.

4.1 Definitions and notation.

4.2 The dihedral groups.

Recommended exercises: 1, 2, 3, 13, 14, 17, 18, 20, 21, 29.

Chapter 5. Cosets and Lagrange's theorem.

5.1 Cosets.

5.2 Lagrange's theorem.

5.3 Fermat's and Euler's theorems.

Recommended exercises: 1, 5, 6, 7, 8, 16, 17, 18, 22, 23, 24.

Chapter 8. Isomorphisms.

8.1 Definition and examples.

8.2 Direct products.

Recommended exercises: 1, 2, 3, 5, 8, 14, 17, 19, 20, 22, 32, 37, 38, 39.

Chapter 9. Homorphisms and factor groups.

9.1 Factor groups and normal subgroups.

9.2 Group homomorphisms.

9.3 The isomorphism theorems.Spolyn

Recommended exercises: 1, 2, 5, 6, 8, 9, 10, 11, 18, 20, 21, 25, 26.

Chapter 11. The structure of groups.

8.1 Finite abelian groups.

Recommended exercises: 1, 2, 6, 7, 8.

Chapter 14. Rings.

14.1 Rings.

14.2 Integral domains and fields.

14.3 Ring homomorphims and ideals.

Recommended exercises: 1, 2, 3, 4, 7, 10, 16, 26, 28, 29, 35, 39.

Chapter 15. Polynomials.

15.1 Polynomial rings.

15.2 The division algorithm.

15.3 Irreducible polynomials.

Recommended exercises: 2, 3, 5, 6, 8, 9, 11, 15, 19, 23, 24, 25.

Chapter 16. Integral domains.

16.1 Fields of fractions.

16.2 Factorization in integral domains.

Recommended exercises: 1, 2, 7, 8, 9, 10, 11, 13, 14, 15, 17, 19.

Chapter 18. Vector spaces.

18.1 Definitions and examples.

18.2 Subspaces.

18.3 Linear independence.

Recommended exercises: 2, 3, 4, 5, 6, 7, 8, 15, 18.

Most of the contents of this section is assumed to be known. Only a short review will be given in the course.

Chapter 19. Fields.

19.1 Extension fields.

19.2 Splitting fields.

19.3 Geometric constructions.

Recommended exercises: 1, 2, 3, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 26.

Chapter 20. Finite fields.

20.1 Structure of a finite field.

Recommended exercises: 1, 2, 3, 5, 6, 7, 8.