In this chapter we investigate two special types of number field in the light of our previous work. The quadratic fields are those of degree 2, and are especially important in the study of quadratic forms. The cyclotomic fields are generated by pth roots of unity, and we consider only the case p prime; these fields are central to Kummer’s approach to Fermat’s Last Theorem and play a substantial role in all subsequent work, including Wiles’s proof. We return to both types of field at later stages. For the moment we content ourselves with finding the rings of integers, integral bases, and discriminants.