Abstract:
For odd prime numbers p, we discuss results on the Galois module
structure
of the p-part of the class group of a cyclic number field of degree p.
In particular one can use genus theory to describe part of this
structure.
This leads to density results for the p-class groups if one considers
the
family of all cyclic number fields of degree p with a fixed number of
primes
dividing the discriminant, and we state a conjecture similar to the
classical
Cohen-Lenstra heuristics for the prime-to-p-part of the class group.
This
conjecture can be proved in some cases, and in addition some numerical
evidence is presented.