Introduction

In this paper we describe an algebraic method to study the structure of (parts of) class groups of abelian number fields. The method goes back to the Hungarian mathematician L. Rédei, who used it to study the 2-primary part of class groups of quadratic number fields in a series of papers [[18]–[24]] that appeared between 1934 and 1953. The case of the l-primary part of the class group of an arbitrary cyclic extension of prime degree l was studied by Inaba [[12], 1940], who realized that one should look at the class group as a module over the group ring. The matter was then taken up by Fröhlich [[6], 1954], who generalized Inaba's results by extending Rédei's quadratic method to the case of a cyclic field of prime power degree. In the seventies, generalizations in the line of Inaba were given by G. Gras [[10]]. In all cases, one studies l-primary parts of the class group of an abelian extension for primes l that divide the degree.

Recently, completely different methods have been developed by Kolyvagin and Rubin, showing that the structure of any l-primary part of the class group of an abelian field of degree coprime to l can be described ‘algebraically’. For primes dividing the degree it is not yet clear whether the approach works. The Kolyvagin-Rubin methods can be seen as refinements of the analytic class number formula, and they are more general than the Rédei-Fröhlich method as they work for most l.