Dirichlet's unit theorem

Gives the rank of the group of units in the ring of algebraic integers of a number field

In mathematics, Dirichlet's unit theorem is a basic result in algebraic number theory due to Peter Gustav Lejeune Dirichlet.[1] It determines the rank of the group of units in the ring OK of algebraic integers of a number field K. The regulator is a positive real number that determines how "dense" the units are.

The statement is that the group of units is finitely generated and has rank (maximal number of multiplicatively independent elements) equal to

As an example, if K is a quadratic field, the rank is 1 if it is a real quadratic field, and 0 if an imaginary quadratic field. The theory for real quadratic fields is essentially the theory of Pell's equation.

Totally real fields are special with respect to units. If L/K is a finite extension of number fields with degree greater than 1 and the units groups for the integers of L and K have the same rank then K is totally real and L is a totally complex quadratic extension. The converse holds too. (An example is K equal to the rationals and L equal to an imaginary quadratic field; both have unit rank 0.)

The theorem not only applies to the maximal order OK but to any order OOK.[2]

The regulator of an algebraic number field of degree greater than 2 is usually quite cumbersome to calculate, though there are now computer algebra packages that can do it in many cases. It is usually much easier to calculate the product hR of the class number h and the regulator using the class number formula, and the main difficulty in calculating the class number of an algebraic number field is usually the calculation of the regulator.

A 'higher' regulator refers to a construction for a function on an algebraic K-group with index n > 1 that plays the same role as the classical regulator does for the group of units, which is a group K1. A theory of such regulators has been in development, with work of Armand Borel and others. Such higher regulators play a role, for example, in the Beilinson conjectures, and are expected to occur in evaluations of certain L-functions at integer values of the argument.[5] See also Beilinson regulator.

The formulation of Stark's conjectures led Harold Stark to define what is now called the Stark regulator, similar to the classical regulator as a determinant of logarithms of units, attached to any Artin representation.[6][7]

Let K be a number field and for each prime P of K above some fixed rational prime p, let UP denote the local units at P and let U1,P denote the subgroup of principal units in UP. Set

Then let E1 denote the set of global units ε that map to U1 via the diagonal embedding of the global units in E.

Since E1 is a finite-index subgroup of the global units, it is an abelian group of rank r1 + r2 − 1. The p-adic regulator is the determinant of the matrix formed by the p-adic logarithms of the generators of this group. Leopoldt's conjecture states that this determinant is non-zero.[8][9]