Singular theorem of ディリクレ

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In mathematics, the singular theorem (Dirichlet's unit theorem) of ディリクレ is a basic result of the algebraic number theory by Peter Gustav ディリクレ (Peter Gustav Dirichlet) [¹]. The singular theorem of ディリクレ decides several single rank (rank) of ring O_K of the algebraic integer of algebra body K. It is an equilateral real number to decide how long regulator (regulator) (or it is said with the singular standard) has "density" of the singular.

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Singular theorem of ディリクレ

As for the singular theorem of ディリクレ, single several groups are limited generation; rank (rank) (the maximum number of causes that are independent for multiplication)

r = r1 + r2-1

I say that に is equal. r₁ is the number of true implantation of K here, and r₂ is the number of conjugate pairs of double bare implantation. As for the characterization of this r₁ and r₂, degree n = [K is based on Q] and じだけあるという way of thinking implantation of K to a complex number body. Because these implantation is one of the implantation to a real number or the implantation to become the pair of the complex conjugate,

n = r1 + 2r₂

となる.

If K is Galois expansion on Q, one of r₁ and r₂ is not 0, but is careful about having possibilities to be 0 with the both sides.

Other methods to decide r₁ and r₂,

• r₁ is a number of the origins of conjugate that is a real number of α when I write it as K = Q(α) using a primitive original theorem, and 2r₂ is a number of the origins of conjugate that is an imaginary number.

• This is the product of the copy of the r₁ unit of R and the copy of C of the r₂ unit when I write tensor product K⊗_QR of the body as the product of the body.

The rank is 1 with the true second body when I assume K the second body as an example, and the rank is 0 with the empty second body. A theory of the true second body is a theory of the Peru equation essentially.

Except Q where a rank is 0 for all several bodies and the empty second body the rank > It is 0. "The size" of the singular is measured generally by the determinant called the regulator (singular standard). The base of the singular can calculate effectively and is practical, and the calculation is very theoretically complicated at n big time.

Several single distortion (torsion) becomes the limited cyclic group with a set of all 1 冪根 of K. In several bodies with at least one true implantation, the distortion should become {1,-1}. Like the empty second body, there are several single several bodies which do not have the true implantation that distortion is {1,-1}.

The total substance is particularly important from the viewpoint of singular. When single several groups of L and single several groups of K do L/K with the same rank as the limited next expansion bigger than 1 a degree, it is the second expansion that is empty with a fruit all-out all-out in L in K. The reverse is also right. (as an example, in the case of the empty second body, several Yuri, L are rank 0 with the both sides K.)

(by future, Claude Chevalley (Claude Chevalley)), the singular theorem were generalized by Helmut Hasse (Helmut Hasse), and the structure of several single group of S-singular (English version) (S-unit) which decided a rank by the localization of the integer ring was described. In addition, it is Galois module structure ${\displaystyle \mathbf {Q} \oplus O_{K,S}\otimes _{\mathbf {Z} }\mathbf {Q}}$  But, it was decided. ^[2]

Regulator (singular standard)

u₁, ...I do it with the meeting of several single generator which speaks, u_r 1, and comes, and assumed a root modulo. If u is an algebraic figure; u¹, ...When assume N_j 1, 2 in response to true implantation double bare implantation as implantation to R and C each in, u^r+1; each element ${\displaystyle N_{j}\log |u_{i}^{j}|}$  As for the line of the である r X (r +1), the friendship of which line has a property to be 0, too (a norm is 1, and all singular is because log of the norm is with the friendship of the element of the line). This means that absolute value R of the determinant of the partial line that is made by deleting the one line does not depend on a line. Numerical value R called regulator (regulator) (or a singular standard) of the algebra body (this value does not depend on the choice of u_i). This value measures "density" of the singular, and it means that there is many singular that a regulator is small.

The regulator has the following geometric interpretation. It is the element of the line in singular u ${\displaystyle N_{j}\log |u^{j}|}$  The representation to pass, and to copy has an image in r dimension subspace of R^r+1, and, from all vectors where the sum becomes 0 of the element, as for the image, it is a lattice in this space by a singular theorem of ディリクレ. The volume of the basic domain of this lattice is R √ (r+1).

In the case of most, the regulator of the algebra body with degree more than 2 has a package of the calculator algebra now, but it is very difficult to usually calculate it. It is easy to calculate product hR which used a regulator in number of the kinds h using a number of the kinds formula, and the main difficulty of the calculation of the number of the kinds of the algebra body is usually to calculate a regulator.

Example

 

A basic domain of several single logarithmic space of the third Japanese yen fission to be provided by adding a root of f(x) = x3 + x2-2x-1 in Q. When α expresses a root of f(x), the set of the basic singular is {ε ₁, ε ₂}. It is ε ₂ = 2 - α ² here in ε ₁ = α ² +α-1. Because the area of the basic domain is approximately 0.910114, the regulator of K is approximately 0.525455.

• The regulator of the empty second body or the regulator of the Yuri integer body is 1. (saying that the determinant of 0*0 line is 1)
• The regulator of the true second body is log of the basic singular. For example, the regulator of Q (√ 5) is log ((√ 5 +1) /2). I do this as follows and understand it. Because the basic singular is /2 (√ 5 +1), and the image of the implantation of two to R is /2 (- √ 5 +1) (√ 5 +1) with /2; the line of r X r +1,

${\displaystyle \left[1\times \log \left|{{\sqrt {5}}+1 \over 2}\right|,\quad 1\times \log \left|{-{\sqrt {5}}+1 \over 2}\right|\ \right]}$ 

である.

• A patrol third body (English version) suffers from the regulator of Q(α) approximately 0.5255 (cyclic cubic field) when I assume α a root of x3 + x2-2x-1. Several single base that assumed べき root modulo is {ε ₁, ε ₂}. It is ε ₁ = α ² +α-1 here and is ε ₂ = 2 - α ². ^[3]

Highly advanced regulator

The highly advanced regulator is n > For 1, it is to constitute the function on the algebraic K-group with the role that a classic singular standard made in single several groups. This is group K₁. The theory of such a regulator develops, and Arman Borel (Armand Borel) and other people study it. Such a regulator plays, for example, an active part in the bay phosphorus loss expectation (Beilinson conjectures), and it is expected what the L-function with with an integer evaluates in a discussion. ^[4]

Stark regulator

By formulation of the Stark expectation, Harold Stark (Harold Stark) proposed a thing called Stark regulator (Stark regulator) now. He proposed Stark regulator as a determinant of log of the singular corresponding to any アルティン expression (English version) (Artin representation) as an analog of the classic regulator. ^[5][6]

p-進 regulator

I assume K several bodies and do it with raw score (prime) P in each fixed Yuri raw score of K and do it when I express local site singular in P in impro_ving it and do it when I express subgroup of the main singular in impro_ving it in U_1,P. Furthermore,

${\displaystyle U_{1}=\prod _{P|p}U_{1,P}}$ 

I do it when I express a set of global singular ε to represent in U₁ through opposite angle implantation of global singular in E in と holder, E₁.

${\displaystyle E_{1}}$  Global single several groups which are a limited index of は 大域的単数 (index) rank it as ${\displaystyle r_{1}+r_{2}-1}$  It is の Abelian group. The p-進 regulator (p-adic regulator) is a determinant of the lines formed of p-進対数 of the generator of this group. In expected (English version) (Leopoldt's conjecture) of Leopold

This article is taken from the Japanese Wikipedia Singular theorem of ディリクレ

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