CM fields with a reciprocal unit-primitive element (Q1704638)

Let \(K\) be an algebraic number field. An element \(\theta\) of \(K\) is called reciprocal if it is conjugated to \(1/\theta\). The authors give first a short proof that \(K\) has a reciprocal primitive element if and only the group of automorphisms of \(K\) contains an involution (Proposition 2.1), and then consider the question of existence of a reciprocal unit generating \(K\). They show that in a totally real field such unit exists if and only if \(\mathrm{Aut}(K)\) has an involution (Proposition 2.2), and characterize non-cyclotomic CM-fields having such units (Theorems 3.3 and 3.4). A part of the last result has been earlier obtained by \textit{F. Lalande} [Problèmes de Galois et nombres algébriques réciproques. Paris: Université Paris-6 (PhD Thesis) (2000)].