Global unit squares and local unit squares
Let \(K\) be any Galois extension of \(\mathbb Q\), and \(U_K\) be the unit group of \(K\). For any place \(v\) of \(K\), let \(U_v\) be the group of local units of \(K_v\). Here the authors study the following problem: does there exist an odd prime \(p\) such that the map \(U_K/U_K^2\to \prod_{v\mid p} U_v/U_v^2\) is injective. They prove that the existence of such primes is equivalent to \(\Hom(U_K/U_K^2, \{\pm 1\})\) is a cyclic \(\mathbb F_2[\text{Gal}(K/\mathbb Q)]\)-module. Moreover, they also prove that if the class number \(h_{\mathbb Q(\zeta_{p^r})^+}\) is odd, then such primes exist for \(\mathbb Q(\zeta_{p^r})^+\) and \(\mathbb Q(\zeta_{p^r})\), where \(p\) is an odd prime and \(\mathbb Q(\zeta_{p^r})^+\) is the maximal real subfield of \(\mathbb Q(\zeta_{p^r})\).
scientific article; zbMATH DE number 5368238