##  [Regulator](/regulator-0) 

 Definition

A positive real invariant of a number field given by the absolute value of the determinant of a matrix whose entries are logarithms of absolute values of a set of fundamental units; it measures the covolume of the unit lattice in the real vector space determined by embeddings.

 

 

 

 

 

 





## Principle

Principle

Dirichlet's unit theorem identifies the free abelian part of the unit group and yields a lattice in a real hyperplane; the regulator is the volume of a fundamental parallelotope for that lattice computed from logarithmic embeddings.

 

 

 

 

 





## Demonstration

Demonstration

For the real quadratic field Q(√2), the fundamental unit is 1+√2; the regulator is |log(1+√2)| (up to the embedding sign convention), giving a concrete one-dimensional covolume equal to the absolute value of that logarithm.

 

 

 

 

## Misapplication

Misapplication

Treating the regulator as an arithmetic count like the class number or using it without accounting for the choice of fundamental units and the embedding-dependent sign conventions, or attempting to define it for fields with finite unit group (e.g. finite fields).

 

 

 

 

 





## Consequence

Consequence

The regulator appears multiplicatively in the analytic class number formula together with the class number and root of unity factor; it quantifies the size of the free part of units and influences distribution of units and heights in Diophantine problems.

 

 

 

 

## Reversal

Reversal

A very small regulator indicates a very dense unit lattice (small covolume) while a very large regulator indicates sparse fundamental units and large gaps in multiplicative structure; zero regulator corresponds to the absence of a free part (finite unit group).

 

 

 

 

 





## Boundary

Boundary

Defined for global fields with infinite unit rank such as number fields; for fields with finite unit group (e.g. finite fields or Q with trivial unit rank) the regulator is either zero or inapplicable; regulator is an archimedean invariant and does not encode p-adic unit sizes.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Close to measures of size such as Mahler measure or heights of units, but conceptually different because it is a lattice covolume derived from logarithms of embeddings and interacts with the class number rather than counting ideals.

 

 

 

 

 





## Synthesis

Synthesis

The regulator is the positive real covolume invariant extracted from the logarithmic embedding of fundamental units; it quantifies the geometric size of the free unit lattice and enters central arithmetic formulae alongside the class number.