This dissertation is a study of two basic questions involving irreducible elements in algebraic number fields. The first question is: Given an algebraic integer β in a field with class number greater than two, how many different lengths of factorizations into irreducibles exist? The distribution into ideal classes of the prime ideals whose product is the principal ideal (β) determines the possible length of the factorizations into irreducibles. Chapter 2 gives precise answers when the field has class number 3 or 4, as well as when the class group is an elementary 2-group of order 8.

The second question is: In a normal extension, when are there rational primes which split completely and remain irreducible? Chapter 3 focusses on the bicyclic bi-quadratic fields. The imaginary bicyclic biquadratic fields which contain such primes are completely determined.