Includes bibliographical references and index.

1. Introduction -- 2. Early history of the Pell equation -- 3. Continued fractions -- 4. Quadratic number fields -- 5. Ideals and continued fractions -- 6. Some special Pell equations -- 7. The ideal class group -- 8. The analytic class number formula -- 9. Some additional analytic results -- 10. Some computational techniques -- 11. (f,p) Representations of O-ideals -- 12. Compact representations -- 13. The subexponential method -- 14. Applications to cryptography -- 15. Unconditional verification of the regulator and the class number -- 16. Principal ideal testing in O -- 17. Conclusion.

Pell's Equation is a very simple Diophantine equation that has been known to mathematicians for over 2000 years. Even today research involving this equation continues to be very active, as can be seen by the publication of at least 150 articles related to this equation over the past decade. However, very few modern books have been published on Pell's Equation, and this will be the first to give a historical development of the equation, as well as to develop the necessary tools for solving the equation. The authors provide a friendly introduction for advanced undergraduates to the delights of algebraic number theory via Pell's Equation. The only prerequisites are a basic knowledge of elementary number theory and abstract algebra. There are also numerous references and notes for those who wish to follow up on various topics.