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 O_K of algebraic integers of a number field K. The regulator is a positive real number that determines how "dense" the units are.

Contents 1 Dirichlet's unit theorem 2 The regulator 3 See also 4 References

[edit] Dirichlet's unit theorem

The statement is that the rank (the maximal number of multiplicatively independent elements) is

r = r₁ + r₂ − 1

where r₁ is the number of real embeddings and r₂ the number of conjugate pairs of complex embeddings of K. This characterisation of r₁ and r₂ 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 = r₁ + 2r₂.

Other ways of determining r₁ and r₂ are

• use the primitive element to write K = Q(α), and then r₁ is the number of conjugates of α that are real, 2r₂ the number that are complex;

• write the tensor product of fields K ⊗_QR as a product of fields, there being r₁ copies of R and r₂ 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 besides Q 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 n is large.

The torsion in the group of units is a finite group of roots of unity, which form a cyclic group. For a number field with at least one real embedding the torsion must therefore be only {1,−1}. There are number fields, for example most imaginary quadratic fields, having no real embeddings which also have {1,−1} for the torsion of its unit group.

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.)

There is a generalisation of the unit theorem by Hasse (and later Chevalley) to describe the structure of the group of S-units, determining the rank of the unit group in localizations of rings of integers.

[edit] The regulator

Suppose that u₁,...,u_r are a set of generators for the unit group modulo roots of unity. If u is an algebraic number, write u¹, ..., u^r+1 for the different embeddings into R or C, and write N_i for the degree of the corresponding embedding over R (so it is 1 for real embeddings and 2 for complex ones). Then the r by r + 1 matrix whose entries are N_ilog|u_i^j| has the property that the sum of any row is zero (because all units have norm 1, and the log of the norm is the sum of the entries of a row). This implies that the absolute value R of the determinant of the submatrix formed by deleting one column is independent of the column. The number R is called the regulator of the algebraic number field (it does not depend on the choice of generators u_i). It measures the "density" of the units: if the regulator is small, this means that there are "lots" of units.

The regulator has the following geometric interpretation. The maps taking a units u to the vector with entries N_ilog|u_i| has image in the r-dimensional subspace of R^r+1 consisting of all vectors whose entries have sum 0, and by Dirichlet's unit theorem the image is a lattice in this subspace. The volume of a fundamental domain of this lattice is R√(r+1).

Examples.

• The regulator of an imaginary quadratic field, or of the rational integers, is 1 (as the determinant of a 0×0 matrix is 1).
• We calculate the regulator of the real quadratic field Q[√5].

A fundamental unit is (√5 + 1)/2, and its images under the two embeddings into R are (√5 + 1)/2 and (−√5 + 1)/2. So the r by r + 1 matrix is

Therefore the regulator is log((√5 +1)/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.

[edit] See also

• Elliptic unit
• cyclotomic unit

[edit] References

• Neukirch Algebraic Number Theory
• Serge Lang, Algebraic number theory, ISBN 0-387-94225-4