Dirichlet's unit theorem explained

In mathematics, Dirichlet's unit theorem is a basic result in algebraic number theory due to Peter Gustav Lejeune Dirichlet. It determines the rank of the group of units in the ring of algebraic integers of a number field . 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 towhere is the number of real embeddings and the number of conjugate pairs of complex embeddings of . This characterisation of and is based on the idea that there will be as many ways to embed in the complex number field as the degree

n=[K:Q]

; these will either be into the real numbers, or pairs of embeddings related by complex conjugation, so that

Note that if is Galois over

then either or .

Other ways of determining and are

use the primitive element theorem to write

K=Q(\alpha)

, and then is the number of conjugates of that are real, the number that are complex; in other words, if is the minimal polynomial of over

, then is the number of real roots and is the number of non-real complex roots of (which come in complex conjugate pairs);

K ⊗ _QR

as a product of fields, there being copies of

and copies of

As an example, if 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.

The rank is positive for all number fields besides

and imaginary quadratic fields, which have rank 0. The 'size' of the units is measured in general by a determinant called the regulator. In principle a basis for the units can be effectively computed; in practice the calculations are quite involved when is large.

The torsion in the group of units is the set of all roots of unity of, which form a finite cyclic group. For a number field with at least one real embedding the torsion must therefore be only