Group units

In Mathematical, and more particularly in Algebra, the Groupe of the units is a notion of the Théorie of the rings

The group of the units is consisted of the whole of the elements of the ring having a Inverse for the second law.

The group of the units is largely used in all the theory of the rings. In the particular case of the ring of the algebraic entireties of a algebraic Bodies of numbers, this group has a well-known structure, thanks to the Théorème of the units of Dirichlet. It forms part, like the Groupe of the classes, the important invariants of the bodies of numbers, and often intervenes in their study, in particular in Cohomologie galoisienne.

Motivation

The properties of an element has of a ring often depend on the principal Idéal has . has associated with this element.

Examples are given by the concepts of irreducible element or element first. The concepts of pgcd or ppcm are also defined starting from an ideal. However if B is another element of the ring such as there exists an invertible element U checking has = U . B , the element B generates the same ideal that has . has . Consequently if has is first, then B is too.

For example, the Polynôme with coefficients in the whole of the rational Nombre S X 2 + 1 divides X 4 - 1, it is the same for 2. X 2 + 2. These polynomials are both irreducible, and thus can belong to the decomposition in factors first of X 4 - 1. The unicity of the decomposition in factors first can be assured only if one associates the two polynomials to regard them only as one single representing.

As far as possible, one uses single representing of a class of associated elements . For example for the polynomials one will add the unit condition to define an irreducible polynomial (i.e. its students' rag procession dominating is equal to one). For Z the whole of the relative whole , a number is known as first if it is positive, one never considers the negative representative, even if there always exists.

Definitions

* an element is known as invertible if, and only, if he admits a reverse for the multiplication. One also speaks about unit of the ring.
* the whole of the invertible elements of a ring is called group of the units or group of invertible the .
* There exists a Relation of equivalence called association relationship defined by: If X and is two elements of the ring there, X is in relation to there and one says X is associated with there if, and only if, there exists an element U group of the unit such as X = U . there .
* There exists relation called relation of division defined by: If X and is there two elements of the ring X is in relation to there and one says X divides there if, and only if, there exists an element has ring such as there = has . X . This relation is compatible with the relation of preceding equivalence and the quotient of the ring by the relation is associated with is a Relation of order.

Demonstrations and fundamental property

The whole of the units is stable for the multiplication, indeed if X and is units there, then there -1. X -1 is the reverse of X . there . The associativeness of the multiplication is guaranteed by the properties of the ring and the existence of a neutral element by the fact that the ring is selected unit. It any more but does not remain to check than each element has a reverse, which is the case by definition of a unit.

The relation is associated with is reflexive because if X is an element of the ring X = 1. X , it is symmetrical because if is an element of the ring there and U a unit such as X = U . there , then there = U -1. X (the ring is supposed to be commutative), finally it is transitive because if X , there and Z is elements of the ring such as there exists U and v in the group of the units with X = U . there and there = v . Z , then X = U . v Z and U . v is invertible. It is thus a Relation of equivalence.

Either X and there two elements of the ring, if X is associated with then X divides there there and divides X there, the association relationship is quite compatible with the relation of division. Moreover if X divides there and if divides X there then there exist two units U and v such as there = U . X and X = v . there what shows that X = U . v . X . As the ring is just U . v is equal to 1 and U and v is units, in conclusion X is associated with there , which shows that the relation of division on the quotient of the ring is antisymmetric. Finally if X divides there and if divides Z there then it exists has and B elements of the ring such as there = has . X and Z = B . there thus Z = has . B . X and the relation is transitive. In conclusion the relation of divisibility is a relation of order on the quotient of the ring by the association relationship .

* the application of the quotient of the ring by the association relationship , provided with the relation of divisibility as a whole of the ideals of the ring provided with the relation with order contains is an isomorphism.

This proposal means two properties: has and B two elements of the ring generate the same ideal, if and only if, has and B is associated, moreover has . has contains B . has if, and only if, has B divides.

Examples

Relative entirety

See also: Whole relative

The group of the units for the ring of the relative entireties is composed of the two elements 1 and -1. The ring is principal, therefore any ideal not no one admits two exactly Antécédent S by the application which with an element has associates has . Z . The two antecedents are has and - has .

To avoid ambiguity, one thus speaks only about the positive representative. Thus a prime number (as the ring is principal, the concept of irreducibility and that of primality is confused and one speaks only prime number in general) in Z is by convention always positive, a pgcd or a PMC is also by definition always positive. This choice makes it possible to obtain without ambiguity a decomposition in factors first single except for a permutation, unlike the case of the positive whole , the decomposition contains in more one factor chosen in the group of the units, either 1 or -1.

Polynomial

See also: Polynomial

If the coefficients of the polynomial are in a body K , then the group of the units is equal to K *, no convention similar to the preceding case does not raise ambiguity.

As previously, the ring is principal, the concepts of polynomial element first and irreducible element is still confused. The tradition forces to use the term of irreducible. A polynomial is known as irreducible if, and only if, any decomposition in two factors contains a unit and if it is not constant.

However any class of equivalence of the association relationship contains a single unit polynomial, i.e. a polynomial whose Monôme dominating is equal to 1. Thus, one in general calls PMC and pgcd the generating unit polynomial of the ideal, thus unicity is still present. In the same way, the theorem of the decomposition in factors first is in general expressed in term of unit polynomial irreducible and unicity with the order of the elements close is restored. This decomposition contains an additional factor element of K *.

In the case or the coefficients of the polynomial are selected in Z , then the group of the unit is equal to {1, -1}. It is of use to take a convention similar to the case of the relative entireties. Thus the irreducible polynomial of a decomposition in factor first, a PMC or a pgcd is selected with positive students' rag procession dominating. This convention is not general.

In the case or the polynomial is with coefficients in an unspecified ring, then no convention does not standardize a canonical representative of a class of association.

Entirety of Gauss

See also: Whole of Gauss

The entireties of Gauss form a Euclidean Anneau, therefore principal. One speaks indifferently about prime number of Gauss or irreducible entirety. The group of the units contains four elements {1, - 1, I, - I}. No particular convention is taken.

Thus an entirety of Gauss and known as irreducible if, and only if any division in two factors contains a unit and that it is not element of the group of the units. 3,-3,3.i and -3.i are called prime numbers of Gauss. If has and B is two entireties of Gauss, then there exist four representatives for the pgcd and the PMC.

The unicity of the decomposition in factors first is expressed with the factors of the group of the unit close .

Algebraic entirety

See also: Whole algebraic

In the general case, the algebraic entireties have only one structure of Anneau of Dedekind, the ring is neither Euclidean neither the factorial main thing nor even . Ambiguity is thus of few consequences and very representative (when there exists) of ideal is considered to have the properties of the ideal. Thus an algebraic entirety is irreducible if, and only if, its ideal is, independently of its representative in the class of association.

The Theorem of the units of Dirichlet watch the existence of several invertible elements in the majority of the algebraic whole rings of . The equality (√5 + 2) (√5 − 2) = 1 is an example.

In the case of a Local ring, this group is easy to describe: it is very exactly complementary to the single one the ideal maximum. The theory of the numbers on a local Corps is some simplified, compared to its total version.

Cyclic ring

The cyclic ring Z / N . Z has for group of the units the whole of the generators of the group. Its cardinal is equal to the Indicatrice of Euler.

If N is first, the ring is the body F p, the group of the units is isomorphous with the cyclic group with N - 1 elements (cf cyclic Groupe).

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