\({\mathbb{Z}{}}_ p\)-independent systems of units (Q1803584)

We quote the author's words: ``Some systems of units known to be independent over \(\mathbb{Z}\) are shown to be independent over some rings of \(p\)-adic integers. The motivation of this study is Leopoldt's conjecture for a finite algebraic extension \(K\) of \(\mathbb{Q}\), which states that for every prime \(p\) the \(\mathbb{Z}_ p\)-rank of the group \(E_ K\) of units (modulo torsion) of \(K\) is equal to the \(\mathbb{Z}\)-rank of \(E_ K\)''.