On the history of the study of ideal class groups

Abstract

This is a survey of a series of results about the class groups of algebraic number fields, with particular emphasis on two articles of Chebotarev [Eine Verallgemeinerung des Minkowski'schen Satzes mit Anwendung auf die Betrachtung der Körperidealklassen, Berichte der wissenschaftlichen Forschungsinstitute in Odessa 1(4) (1924) 17–20; Zur Gruppentheorie des Klassenkörpers, J. Reine Angew. Math. 161 (1929/30) 179–193; corrigendum, ibid. 164 (1931) 196] which seem to be almost forgotten. Their relationship to earlier work on the one hand, and to selected subsequent contributions on the other hand, is discussed. In this way, there emerges an interesting line of development, up to the present day, of results due to Kummer, Hasse, Leopoldt, Iwasawa, and others. More recent work treated here includes results by Cornell and Rosen (1981) and Lemmermeyer (2003) describing the structure of the class group under quite general conditions.