Dirichlet's unit theorem

Dirichlet's unit theorem 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 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 {displaystyle n=[K:mathbb {Q} ]} ; these will either be into the real numbers, or pairs of embeddings related by complex conjugation, so that n = r1 + 2r2.

Note that if K is Galois over {Anzeigestil mathbb {Q} } then either r1 = 0 or r2 = 0.

Other ways of determining r1 and r2 are use the primitive element theorem to write {displaystyle K=mathbb {Q} (Alpha )} , and then r1 is the number of conjugates of α that are real, 2r2 the number that are complex; mit anderen Worten, if f is the minimal polynomial of α over {Anzeigestil mathbb {Q} } , then r1 is the number of real roots and 2r2 is the number of non-real complex roots of f (which come in complex conjugate pairs); write the tensor product of fields {displaystyle Kotimes _{mathbb {Q} }mathbb {R} } as a product of fields, there being r1 copies of {Anzeigestil mathbb {R} } and r2 copies of {Anzeigestil mathbb {C} } .

Als Beispiel, if K is a quadratic field, the rank is 1 if it is a real quadratic field, und 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 {Anzeigestil mathbb {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 the set of all roots of unity of K, which form a finite 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.) The theorem not only applies to the maximal order OK but to any order O ⊂ OK.[2] There is a generalisation of the unit theorem by Helmut Hasse (and later Claude Chevalley) to describe the structure of the group of S-units, determining the rank of the unit group in localizations of rings of integers. Ebenfalls, the Galois module structure of {Anzeigestil mathbb {Q} oplus O_{K,S}omal _{mathbb {Z} }mathbb {Q} } has been determined.[3] Inhalt 1 The regulator 1.1 Beispiele 2 Higher regulators 3 Stark regulator 4 p-adic regulator 5 Siehe auch 6 Anmerkungen 7 References The regulator Suppose that K is a number field and {displaystyle u_{1},Punkte ,u_{r}} are a set of generators for the unit group of K modulo roots of unity. There will be r + 1 Archimedean places of K, either real or complex. Zum {displaystyle uin K} , write {Anzeigestil u^{(1)},Punkte ,u^{(r+1)}} for the different embeddings into {Anzeigestil mathbb {R} } oder {Anzeigestil mathbb {C} } and set Nj to 1 oder 2 if the corresponding embedding is real or complex respectively. Then the r × (r + 1) Matrix {Anzeigestil links(N_{j}Protokoll übrig|u_{ich}^{(j)}Rechts|Rechts)_{i=1,dots ,r,;j=1,dots ,r+1}} 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 in 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 ui). It measures the "Dichte" of the units: if the regulator is small, this means that there are "lots" of units.

The regulator has the following geometric interpretation. The map taking a unit u to the vector with entries {textstyle N_{j}Protokoll übrig|u^{(j)}Rechts|} has an image in the r-dimensional subspace of {Anzeigestil mathbb {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 {Anzeigestil R{quadrat {r+1}}} .

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.