Editing Dirichlet's unit theorem (section)

=={{mvar|p}}-adic regulator==
Let {{mvar|K}} be a [[number field]] and for each [[Valuation (algebra)|prime]] {{mvar|P}} of {{mvar|K}} above some fixed rational prime {{mvar|p}}, let {{math|''U''<sub>''P''</sub>}} denote the local units at {{mvar|P}} and let {{math|''U''<sub>1,''P''</sub>}} denote the subgroup of principal units in {{math|''U''<sub>''P''</sub>}}. Set <math display="block"> U_1 = \prod_{P|p} U_{1,P}. </math>

Then let {{math|''E''<sub>1</sub>}} denote the set of global units {{mvar|ε}} that map to {{math|''U''<sub>1</sub>}} via the diagonal embedding of the global units in {{mvar|E}}.

Since {{math|''E''<sub>1</sub>}} is a finite-[[Index of a subgroup|index]] subgroup of the global units, it is an [[abelian group]] of rank {{math|''r''<sub>1</sub> + ''r''<sub>2</sub> − 1}}. The '''{{mvar|p}}-adic regulator''' is the determinant of the matrix formed by the {{mvar|p}}-adic logarithms of the generators of this group. ''[[Leopoldt's conjecture]]'' states that this determinant is non-zero.<ref name=NSW6267>Neukirch et al. (2008) p. 626–627</ref><ref>{{cite book | last=Iwasawa | first=Kenkichi | author-link=Kenkichi Iwasawa | title=Lectures on {{mvar|p}}-adic {{mvar|L}}-functions | series=Annals of Mathematics Studies | volume=74 | location=Princeton, NJ | publisher=Princeton University Press and University of Tokyo Press | year=1972 | isbn=0-691-08112-3 | zbl=0236.12001 | pages=36–42 }}</ref>