Introduction 1. Integral domains 2. Euclidean domains 3. Noetherian domains 4. Elements integral over a domain 5. Algebraic extensions of a field 6. Algebraic number fields 7. Integral bases 8. Dedekind domains 9. Norms of ideals 10. Decomposing primes in a number field 11. Units in real quadratic fields 12. The ideal class group 13. Dirichlet's unit theorem 14. Applications to diophantine equations.