Different groups of circular units of a compositum of real quadratic fields

There are many different definitions of the group of circular units of a real abelian field. The aim of this paper is to study their relations in the special case of a compositum k of real quadratic fields such that -1 is not a square in the genus field K of k in the narrow sense. The reason why fields of this type are considered is as follows. In such a field it is possible to define a group C of units (slightly bigger than Sinnott's group of circular units) such that the Galois group acts on C/(±C²) trivially (see [K, Lemma 2]). Due to this key property we can easily compare different groups of circular units (see the conclusion of this paper).