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Let  $  F $
be a totally real algebraic number field (cf. also [[Field|Field]]; [[Algebraic number|Algebraic number]]) and let  $  p $
be a prime number. Let  $  {\sigma _ {1} \dots \sigma _ {r _ {1}  } } : F \rightarrow {\mathbf C _ {p} } $
denote the distinct embeddings of  $  F $
into the completion  $  \mathbf C _ {p} $
of the algebraic closure of  $  \mathbf Q _ {p} $. 
By the Dirichlet unit theorem (cf. also [[Dirichlet theorem|Dirichlet theorem]]), the unit group  $  U _ {F} $
of  $  F $
has rank  $  r = r _ {1} - 1 $. 
Let  $  \epsilon _ {1} \dots \epsilon _ {r} $
be a  $  \mathbf Z $-
basis of  $  U _ {F} $. 
In [[#References|[a5]]], H.-W. Leopoldt defined the  $  p $-
adic regulator  $  R _ {p} ( F ) $
as the  $  p $-
adic analogue of the Dirichlet regulator:

$$ 
R _ {p} ( F ) = \pm  { \mathop{\rm det} } \left ( { \mathop{\rm log} } _ {p} ( \sigma _ {i} ( \epsilon _ {j} ) ) _ {1 \leq  i,j \leq  r }  \right ) ,
$$

where  $  { { \mathop{\rm log} } _ {p} } : {U _ {F} } \rightarrow {\mathbf C _ {p} } $
denotes the  $  p $-
adic logarithm.

Leopoldt's conjecture is:  $  R _ {p} ( F ) \neq 0 $.

The definition of  $  R _ {p} ( F ) $(
and therefore also the conjecture) extends to arbitrary number fields (cf. [[#References|[a7]]]) and is nowadays considered in this generality. A. Brumer used transcendental methods developed by A. Baker to prove Leopoldt's conjecture for fields  $  F $
that are Abelian over  $  \mathbf Q $
or over an imaginary quadratic field [[#References|[a2]]]. For specific non-Abelian fields the conjecture has also been verified (cf., e.g., [[#References|[a1]]]), but in general it is still (1996) open.

For a totally real field  $  F $, 
Leopoldt's conjecture is equivalent to the non-vanishing of the  $  p $-
adic  $  \zeta $-
function  $  \zeta _ {F,p }  ( s ) $
at  $  s = 1 $(
cf. [[#References|[a5]]], [[#References|[a3]]]).

For a prime  $  v $
in  $  F $, 
let  $  U _ {v} $
denote the group of units of the local field  $  F _ {v} $. 
There is a canonical mapping

$$ 
{f _ {p} } : {U _ {F} \otimes \mathbf Z _ {p} } \rightarrow {\prod _ { {v \mid  p } } U _ {v} }
$$

and the Leopoldt defect  $  \delta _ {F} $
is defined as the  $  \mathbf Z _ {p} $-
rank of  $  { \mathop{\rm ker} } f _ {p} $. 
[[Class field theory|Class field theory]] yields the following equivalent formulation of the Leopoldt conjecture (cf. [[#References|[a7]]]): Leopoldt's conjecture holds if and only if  $  \delta _ {F} = 0 $.

==Relation to Iwasawa theory.==
An extension  $  F _  \infty  /F $
of a number field  $  F $
is called a  $  \mathbf Z _ {p} $-
extension if it is a [[Galois extension|Galois extension]] and  $  { \mathop{\rm Gal} } ( F _  \infty  /F ) \cong \mathbf Z _ {p} $. 
The number of independent  $  \mathbf Z _ {p} $-
extensions of  $  F $
is related via class field theory to the  $  \mathbf Z _ {p} $-
rank of  $  { \mathop{\rm coker} } f _ {p} $
and is equal to  $  1 + r _ {2} ( F ) + \delta _ {F} $(
cf. [[#References|[a4]]]), where  $  r _ {2} ( F ) $
is the number of pairs of complex-conjugate embeddings of  $  F $.

For  $  n \geq  0 $, 
let  $  F _ {n} $
denote the unique subfield of  $  F _  \infty  /F $
of degree  $  p  ^ {n} $
over  $  F $
and let  $  \delta _ {n} $
denote the Leopoldt defect of  $  F _ {n} $. 
The  $  \mathbf Z _ {p} $-
extension  $  F _  \infty  /F $
satisfies the weak Leopoldt conjecture if the defects  $  \delta _ {n} $
are bounded independent of  $  n $. 
It is known (cf. [[#References|[a4]]]) that the weak Leopoldt conjecture holds for the so-called cyclotomic  $  \mathbf Z _ {p} $-
extension of  $  F $, 
i.e. for the unique  $  \mathbf Z _ {p} $-
extension contained in  $  F ( \mu _ {p  ^  \infty   } ) $.

==Relation to Galois cohomology.==
Let  $  G _ {p} ( F ) $
denote the [[Galois group|Galois group]] of the maximal pro- $  p $-
extension of  $  F $, 
which is unramified outside  $  p $. 
Leopoldt's conjecture is equivalent to the vanishing of the [[Galois cohomology|Galois cohomology]] group  $  H  ^ {2} ( G _ {p} ( F ) , \mathbf Q _ {p} / \mathbf Z _ {p} ) $[[#References|[a6]]]. More generally, it is conjectured that

$$ 
H  ^ {2} ( G _ {p} ( F ) , \mathbf Q _ {p} / \mathbf Z _ {p} ( i ) ) = 0
$$

for all  $  i \neq 1 $[[#References|[a6]]]. This is known to be true for  $  i \geq  2 $
as a consequence of a profound result of A. Borel in [[Algebraic K-theory|algebraic  $  K $-
theory]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> F. Bertrandias, J.-J. Payan, "$\Gamma$-extensions et invariants cyclotomiques" ''Ann. Sci. Ecole Norm. Sup. (4)'' , '''5''' (1972) pp. 517–543 {{MR|0480419}} {{MR|0337882}} {{ZBL|0246.12005}} {{ZBL|0246.12004}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> A. Brumer, "On the units of algebraic number fields" ''Mathematica'' , '''14''' (1967) pp. 121–124 {{MR|0220694}} {{ZBL|0171.01105}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> P. Colmez, "Résidu en $s=1$ des fonctions zêta $p$-adiques" ''Invent. Math.'' , '''91''' (1988) pp. 371–389</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> K. Iwasawa, "On $\ZZ_\ell$-extensions of algebraic number fields" ''Ann. of Math.'' , '''98''' (1973) pp. 246–326 {{MR|349627}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> H.-W. Leopoldt, "Zur Arithmetik in abelschen Zahlkörpern" ''J. Reine Angew. Math.'' , '''209''' (1962) pp. 54–71 {{MR|0139602}} {{ZBL|0204.07101}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> P. Schneider, "Über gewisse Galoiscohomologiegruppen" ''Math. Z.'' , '''168''' (1979) pp. 181–205 {{MR|0544704}} {{ZBL|0421.12024}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> L.C. Washington, "Introduction to cyclotomic fields" , Springer (1982) {{MR|0718674}} {{ZBL|0484.12001}} </TD></TR></table>