The Existence Theorem and Local Class Field Theory

Abstract

We must prove that any open subgroup H of J _k containing k* belongs to some abelian extension. Thus at some point, we have to start exhibiting abelian extensions of k. There are not that many ways of doing this. One general way is to make cyclotomic extensions, and when the n-th roots of unity are in k, to make Kummer extensions, i.e. adjoining n-th roots of elements of k. We shall prove the existence theorem by this method. Deeper methods involving the values of certain transcendental functions are more significant, but lead into directions which require a whole book to themselves. We first start with the reduction lemma.