One usual "proof" of Leopoldt Conjecture is that $\mathbb{Z}_p$ is $\mathbb{Z}$-flat, hence the rank of the $p$-adic completion of the units of a number field has the same rank of the units themselves (which is Leopoldt Conjecture) because you can obtain the completion simply as $\mathcal{O}^times\otimes\mathbb{Z}_p$. \mathcal{O}^\times\otimes\mathbb{Z}_p$. 1 [made Community Wiki] One usual "proof" of Leopoldt Conjecture is that$\mathbb{Z}_p$is$\mathbb{Z}$-flat, hence the rank of the$p$-adic completion of the units of a number field has the same rank of the units themselves (which is Leopoldt Conjecture) because you can obtain the completion simply as$\mathcal{O}^times\otimes\mathbb{Z}_p\$.