ABHYANKAR'S Lemma provides a siniple method of producing unramified extensions of number fields. By application of the lemma we simultaneously simplify and strengthen some recent results of ISHIDA concerning unramified abelian extensions and class numbers.
$ 1 . The main object of this note is to show that the results obtained by ISHIDA in a series of papers ([4], [ 5 ] , [S]) can be deduced more easily by the use of ABHYANKAR'S lemma. I n fact, we show how to strengthen and generalize ISHIDA'B theorems by this method. The applicability of ABHYANKAR'S lemma t o some of these results has already been observed in 191, [2], and [S].
Let F be a field of algebraic number which is a finite extension of Q , the field of rational numbers. The maximal abelian unramified extension P of F is called its HILBERT class field. By ARTIN'S reciprocity law, Gal ( P / F ) is isomorphic to the ideal class group of F . Because of the importance of class groups in number theory, it is interesting to ask: Given an algebraic number field F , how does one construct abelian unrarnified extensions KIP? For a given F, one can use cyclotomic field to create extensions the ramification of which is absorbed in F/Q by ABHYANKAR'S lemma. This was done in [9]. Frey and GEYER [2] used this idea t o construct unramified extensions of a wider class of fields. They also used a method suggested by E. ARTIN, (see [7], p. l 2 l ) , for constructing unramified extensions using the approximation theorem.