Explicit elements of norm one for cyclic groups

Ginosar and the first author reduced the surjectivity of the norm map NG for a group G to the surjectivity of the norm maps for its elementary abelian subgroups; more precisely, they proved that NG : R → R G is surjective if and only if NU : R → R U is surjective for every elementary abelian subgroup U of G (see [2, Theorem 1]). The R -linearity of NU implies that it is surjective if and only there exists an element xU ∈ R such that NU (xU ) = 1. Suppose we have such an element xU for every elementary abelian subgroup U of G. Then by the result mentioned above there is a “global” element xG ∈ R such that NG(xG) = 1. Using this last statement, Shelah observed (see [2, Proposition 6]) that there exists a formula in which xG is a finite sum of the form