Note on an Ordering Theorem for Subfields

Extract

In a recent paper [3] Tannaka gave an interesting ordering theorem for subfields of a p-adic number field, The purpose of the present note is firstly to observe, on modifying Tannaka’s argument a little, that his restriction to those subfields over which the original field is abelian may be removed and in fact the theorem holds for arbitrary fields which are not p-adic number fields, indeed in a much refined form, and secondly to formulate a similar ordering theorem for algebraic number fields in terms of idèle-class groups in place of element groups.