ABSTRACT
The results of this chapter, although apparently diverse, all have a strong bearing on the question of practical computation of the class-number, within the limits of the techniques now at our command. In the first section we study a special case of how a rational prime breaks up into prime ideals in a number field. The second section supplements this by showing that the distinct classes of fractional ideals may be found from the prime ideals dividing a finite set of rational primes, this set being in some sense `small' provided the degree of K and its discriminant are not too 'large'. Several specific cases are studied, especially quadratic fields: in particular we complete the list of fields Q(i) with negative d and with class-number 1 (although we do not prove our list complete).
