Remark on Polický's paper on circular units of a compositum of quadratic number fields (Q555280)

The term ``circular unit'' encompasses a range of definitions of global unit, typically in an abelian number field. What the various definitions have in common is that units of this type have connections with the class-group of the field. For example, Sinnott's group of circular units in the maximal real subfield of a cyclotomic field has (finite) index given by a formula involving the class number and an explicit power of 2. To give an impression of what circular units look like, they are constructed from elements of the form \(1 -\zeta\) where \(\zeta\) is a root of unity in the field. The setting of the paper under review is of a finite compositum \(k\) of quadratic fields, together with its narrow genus field \(K\). Assume that neither \(\sqrt{-1}\) nor \(\sqrt 2\) is in \(K\). In this situation, \textit{Z. Polický} defined in [``On the index of circular units in the full group of units in a compositum of quadratic fields'', J. Number Theory 128, No. 4, 1074--1090 (2008; Zbl 1204.11171)] a group \(C\) of circular units in \(k\) that contains Sinnott's group, and further specified five criteria C1--C5 with the property that given \(\varepsilon\in C\), the following are equivalent: (a) \(\varepsilon\) or \(2\varepsilon\) is a square in \(K\). (b) C1--C5 all hold. We give a flavour of the criteria C1--C5: For each \(\sigma\in G = \text{Gal}(K/\mathbb Q)\), \(\varepsilon^{1-\sigma}\) factors in a certain way into uniquely determined roots of unity of order 1, 3 or 4. The criteria take the form of certain relations among these roots of unity. The main aim of the article is to show that in fact C1 implies C2 and C3, and so only C1, C4 and C5 are required.