Just suppose, for a moment, that all rings of integers in algebraic
number fields were unique factorization domains, then it would be fairly
easy to produce a proof of Fermat's Last Theorem, fitting, say, in the
margin of this page. Unfortunately however, rings of integers are not
that nice in general, so that, for centuries, math- ematicians had to
search for alternative proofs, a quest which culminated finally in
Wiles' marvelous results - but this is history. The fact remains that
modern algebraic number theory really started off with in- vestigating
the problem which rings of integers actually are unique factorization
domains. The best approach to this question is, of course, through the
general the- ory of Dedekind rings, using the full power of their class
group, whose vanishing is, by its very definition, equivalent to the
unique factorization property. Using the fact that a Dedekind ring is
essentially just a one-dimensional global version of discrete valuation
rings, one easily verifies that the class group of a Dedekind ring
coincides with its Picard group, thus making it into a nice, functorial
invariant, which may be studied and calculated through algebraic,
geometric and co homological methods. In view of the success of the use
of the class group within the framework of Dedekind rings, one may
wonder whether it may be applied in other contexts as well. However, for
more general rings, even the definition of the class group itself causes
problems.
