Early History of the Pell Equation.- Continued Fractions.- Quadratic Number Fields.- Ideals and Continued Fractions.- Some Special Pell Equations.- The Ideal Class Group.- The Analytic Class Number Formula.- Some Additional Analytic Results.- Some Computational Techniques.- (f, p) Representations of -ideals.- Compact Representations.- The Subexponential Method.- Applications to Cryptography.- Unconditional Verification of the Regulator and the Class Number.- Principal Ideal Testing in .- Conclusion.