Regulator of an algebraic number field

$ K $

The number $ R _ {K} $ that is, by definition, equal to 1 if $ K $ is the field $ \mathbf Q $ or an imaginary quadratic extension of $ \mathbf Q $, and to $ v/ \sqrt {r+1} $ in all other cases, where $ r $ is the rank of the group $ E $ of units of the field $ K $( see Algebraic number; Algebraic number theory) and $ v $ is the $ r $- dimensional volume of the basic parallelepipedon of the $ r $- dimensional lattice in $ \mathbf R ^ {r+1} $ that is the image of $ E $ under its logarithmic mapping $ l $ into $ \mathbf R ^ {r+1} $. The homomorphism $ l $ is defined as follows: Let $ \sigma _ {1} \dots \sigma _ {s} $ be all real and let $ \sigma _ {s+1} \dots \sigma _ {s+t} $ be all pairwise complex non-conjugate isomorphisms of $ K $ into $ \mathbf C $; $ s + 2t = \mathop{\rm dim} _ {\mathbf Q} K $. Then $ {r+1} = {s+t} $( see Dirichlet theorem on units), and $ l: E \rightarrow \mathbf R ^ {r+1} $ is defined by the formula

$$ l( \alpha ) = ( l _ {1} ( \alpha ) \dots l _ {s+t} ( \alpha )), $$

where

$$ l _ {i} ( \alpha ) = \left \{ \begin{array}{ll} \mathop{\rm ln} | \sigma _ {i} ( \alpha ) | &\textrm{ if } 1 \leq i \leq s, \\ \mathop{\rm ln} | \sigma _ {i} ( \alpha ) | ^ {2} &\textrm{ if } {s+1} \leq i \leq {s+t}. \\ \end{array} \right . $$

The image of $ E $ under $ l $ is an $ r $- dimensional lattice in $ \mathbf R ^ {r+1} $ lying in the plane $\sum_{i=0}^ {r+1} x _ {i} = 0$ (where the $x_i$ are the canonical coordinates).

Units $ \epsilon _ {1} \dots \epsilon _ {r} $ for which $ l( e _ {1} ) \dots l( e _ {r} ) $ form a basis of the lattice $ l( E) $ are known as fundamental units of $ K $, and

$$ R _ {K} = \| \mathop{\rm det} ( l _ {i} ( \epsilon _ {j} )) _ {i,j=1} ^ {r} \| . $$

There are other formulas linking the regulator with other invariants of the field $ K $( see, for example, Discriminant, 3).

If instead of $ E $ one considers the intersection of this group with an order $ {\mathcal O} $ of $ K $, then the regulator $ R _ {\mathcal O} $ of $ {\mathcal O} $ can be defined in the same way.

References

[1]	Z.I. Borevich, I.R. Shafarevich, "Number theory" , Acad. Press (1987) (Translated from Russian) (German translation: Birkhäuser, 1966)
[2]	S. Lang, "Algebraic number theory" , Addison-Wesley (1970)