A remark on class numbers of number field extensions

In this note we study some cases in which the structure of the Galois group of an extension of number fields gives information about the relation between the ideal class groups of these fields.The letters K, L, etc. will denote finite extension fields of the rationals and the letter p will denote a rational prime.If A is an abelian group we let its order be denoted by |^4| and its p-sylow subgroup by Ap.The class group of L will be written Cl¿.Lemma 1.Let S be a set of automorphisms of L, satisfying <r¿¿l, (T2= 1, for all aÇES.Let H be the group of products of even numbers of elements of S. Then if p does not divide the class number of the fixed field of any of the <r£S, the action of H on (Cl¿)p is trivial.Proof.It will suffice to show that (C)" = C-\ vGS, CG(Ch)P-Let I be an ideal of L representing C. Then for some », 7P" = (a), the principal ideal generated by a £ L. Now 77* = /, the extension to L of an ideal of the fixed field of <r.Hence J"" = (aa')pn = (b), b in the fixed field of <r.But from our assumption on the class number of this field it follows that J = (c), c in L, hence CC" = l. Corollary.Let L, S, 77, p be as above, let M be the maximum unramified abelian p-extension of L, and let K be the fixed field of 77.Then