Dirichlet's unit theorem
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In algebraic number theory, Dirichlet's unit theorem determines the rank of the group of units in the ring OK of algebraic integers of a number field K. The statement is that the rank is
- r + s − 1
where r is the number of real embeddings and 2s the number of complex embeddings of K. This characterisation of r and s is based on the idea that there will be as many ways to embed K 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
- n = r + 2s.
Other ways of determining r and s are
- use the primitive element to write K = Q(α), and then r is the number of conjugates of α that are real, 2s the number that are complex;
- write the tensor product of fields K ⊗QR as a product of fields, there being r copies of R and s copies of C.
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.
The rank is > 0 for all number fields other than Q and imaginary quadratic fields. 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 n is large.
The torsion in the group of units is always a cyclic group generated by some root of unity. For a totally real field the torsion must therefore be only {1,−1}.
There is a generalisation of the unit theorem to so-called S-units, determining the rank of the unit group in localizations of rings of integers.