The p-adic regulator of a number field does not vanish.
~Formally~
Let \(K\) be a number field and for each prime \(P\) of \(K\) above some fixed rational prime \(p\), let \(U_P\) denote the local units at \(P\) and let \(U_{1,P}\) denote the subgroup of principal units in \(U_P\) and set $$U_1=\prod_{P|p}{U_{1,P}}$$ and let \(E_1\) denote the set of global units \(\epsilon\) that map \(U_1\) via the diagonal embedding of the global units in \(E\). Then the \( \mathbb{Z}_p \)-module rank of the closure of \(E_1\) embedded diagonally in \(U_1\) is \(r_1+r_2-1\) where \(r_1\) is the number of real embeddings of \(K\) and \(r_2\) is the number of pairs of complex embeddings.
Note: \(E_1\) is a finite-index subgroup of the global units, and is thus an abelian group of rank \(r_1+r_2-1\).
| Author | Date Published | |
| Heinrich-Wolfgang Leopoldt [1] | 1962 |
[1] Leopoldt, Heinrich-Wolfgang (1962), "Zur Arithmetik in abelschen Zahlkörpern", Journal für die reine und angewandte Mathematik, 209: 54–71, ISSN 0075-4102, MR 0139602, Zbl 0204.07101
number field; diagonal embedding; real embeddings; complex embeddings; finite-index subgroup