On the Washington subgroup of the unit group of certain abelian fields

Abstract

For an abelian number field K, Sinnott defined the group C _S (K) of circular units of K and found the index formula for [E(K):C _S (K)], where E(K) is the group of units of K. Let C _W (K) be the subgroup of E(K) consisting of the cyclotomic units of $\mathbb{Q} (\zeta_{n} )$ fixed by $\mathrm{Gal}(\mathbb{Q}(\zeta_{n})/K)$ as was suggested by Washington. The aim of this paper is to find the index formula for [E(K):C _W (K)] when the conductor of K has two distinct odd prime divisors. To do this, we first exhibit a basis of C _W (K), and then describe the group structure of C _W (K)/C _S (K) by comparing the basis of C _W (K) with that of C _S (K) given by Dohmae, from which the index formula for [E(K):C _W (K)] can be derived.