We start from the question:
When are the irreducible integers of a number field determined by their norms?
Attention is centered on the case in which the word “norm” is taken to mean the
relative norm of a Galois extension. In this case we are able to show that the ideal
class group, as a module over the Galois group, is severely limited by this condition.
The restriction of this question to (relatively) quadratic extensions has special
properties which are studied in further detail. The homological methods which are in
the background of our study become very useful in the study of quadratic
extensions.
