Euclidean ring

In Mathematical and more precisely in Algebra, within the framework of the Theory of the rings, a Euclidean ring is a particular type of commutative ring unit integrates.

A ring is known as Euclidean if, and only if, it is possible to define a Euclidean Division.

This property is rich consequences, an Euclidean ring is always principal, it checks the Identité of Bézout, the Lemme of Euclide, it is factorial and satisfies the conditions of the fundamental Théorème of arithmetic the.

Such a structure is rich in theorems because it is possible to build there a Arithmétique and particularly a modular Arithmétique.

History

Origin

The first reference having influenced the mathematical world on the question of Euclidean division is book VII, of the Éléments of Euclide going back to approximately 300 years before J. - C. The first construction theoretical of division and the study of his consequences are found there. This branch of mathematics takes the name of Arithmétique.

Certain mathematicians like Diophante of Alexandria (approx. 200/214 - approx. 284/298) or Pierre de Fermat (1601 - 1665) include/understand the extraordinary richness of this branch of mathematics. They establish some results like the Petit theorem of Fermat and formulate conjectures like the Théorème of the two squares of Fermat or the Grand theorem of Fermat. At the 18th century, some are shown. One can quote Leonhard Euler (1707 - 1783) with the theorem of the two squares, or the case N equal to three of the great theorem of fermat, almost treated in 1753. Other conjectures as that of the quadratic Loi of reciprocity appear. These results are essentially shown thanks to a remarkable virtuosity on behalf of the mathematicians, but the theoretical contribution is weak, consequently the results are not very generalizable. There exists nevertheless a notable exception with the quadratic forms of Joseph-Louis Lagrange (1736 - 1813) in 1775.

Emergence of the concept

In 1801 Carl Friedrich Gauss (1777 - 1855) discovers the first ring of algebraic numbers, beyond that of the integers, that of the whole which bear its name now. This ring has an Euclidean division, and consequently, the Identité of Bézout, the Lemme of Euclide, and especially the fundamental Théorème of arithmetic the apply. In the same way, the ring of the polynomials with coefficients in a commutative body, also has an Euclidean division. Gauss thus built there an arithmetic analog with the preceding ones.

This approach is used for others whole algebraic, for example by Ferdinand Eisenstein (1823 - 1852) which discovers the whole of the numbers called whole of Eisenstein and which has an Euclidean division.

The contribution of an Euclidean division to a structure is a fertile step. Gauss makes use of it for one of its demonstrations of the quadratic law of reciprocity, of tangible progress are carried out for the resolution of the great theorem of Fermat. The case N equal to three becomes perfectly rigorous. The cases N equal to five, then fourteen, then seven, are shown, with the massive contribution of other ideas. The application of the decomposition in factors first to the cyclotomic polynomials makes it possible Gauss to find new a regular Polygone constructible with the rule and the compass, the heptadécagone or regular polygon with seventeen with dimensions.

The idea is so innovative and profitable that the lemma of Euclide and the fundamental theorem of arithmetic are sometimes renamed lemma of Gauss and theorem of Gauss . The book of arithmetic of Gauss is worth quickly with its author, the nickname of prince of the mathematicians .

Formalization

Paradoxically modern formalization comes from the limitations of the arithmetic preceding ones. By a step using the concept of ring of entireties, Gabriel Lamé (1795 - 1870) thinks that it showed the great theorem of Fermat. Ernst Kummer (1810 - 1893) watch for an example in 1844 that a ring of entireties does not lay out, in general, of a single decomposition in factors first. This result invalidates the proof of Lamé. Kummer discovers in 1846 a new concept which he baptizes ideal complex number to find in a new form unicity necessary.

This work opens the way with the formalization of the structure of ring. One can quote Richard Dedekind (1831 1916) and David Hilbert (1862 1943) among the principal contributors. An Euclidean ring becomes a simple particular case of a vast theory, the Théorie of the rings.

Examples

Whole relative

See also: Euclidean Division

The relative entireties form the prototype of the Euclidean ring:

There one recognizes the form of Euclidean division in the whole of the natural whole $\backslash \; mathbb\; \{NR\}$ for which | N | = N . One can notice however that, on the one hand $\backslash \; mathbb\; \{NR\}$ is not a ring, on the other hand it is not specified here the unicity of Q and R . This is explained by the fact that, to be able to prolong with $\backslash \; mathbb\; \{Z\}$ (together of the relative whole ) the definition of division in $\backslash \; mathbb\; \{NR\}$, it is necessary, or to fix an additional condition on B ( B > 0) thus restricting the field of validity of Euclidean division, or to accept to take B negative and to take for definition has = bq + R with | R | < | B |. But then one can find two decompositions possible:

19 = (- 5) × (- 3) + 4 with |4| < |- 5| but also 19 = (- 5) × (- 4) + (- 1) with |- 1| < |- 5|

This division makes it possible to build a Arithmétique checking the following properties:

* the ideal of the entireties are the whole of multiples (form $n\; \backslash \; mathbb\; \{Z\}$). The ring is known as principal.

* the Identité of Bézout is checked.

* the Lemme of Euclide is checked.

* the fundamental Théorème of arithmetic the applies.

Consequently, it is possible to define: the family of the prime numbers, PC as well as the pgcd. The Anneau quotient $\backslash \; mathbb\; \{Z\}\; /n\; \backslash \; mathbb\; \{Z\}$ is well defined, it is the structure at the base of the modular Arithmétique.

The first known application is probably the demonstration of the irrationality of the square root of two. The Petit theorem of Fermat is quickly shown once established the fact that if N is first $\backslash \; mathbb\; \{Z\}\; /n\; \backslash \; mathbb\; \{Z\}$ has a structure of body. Fermat largely uses this arithmetic, for example to show the absence of solution for its great theorem if N is equal to four. Euler gives a broad quantity of examples of use of arithmetic in $\backslash \; mathbb\; \{Z\}$, like the exhaustive study of the equation of Pell.

These results are the properties which justified the creation of the abstract concept of Euclidean ring. Indeed, all these properties are the consequences only of only one, Euclidean division.

Polynomial S with coefficients in a Commutative body

See also: Polynomial and Euclidean division

If a body K is commutative, then the ring of the polynomials K is Euclidean. Division takes the following form:

If the form is overall similar to that of the entireties, it is noticed nevertheless that a relation of order on the unit K is not necessary. It is enough to an application, similar to that which, with a polynomial associates its degree, and whose whole of arrival is ordered.

The arithmetic one is based on the same consequences, the ring is principal, the identity of Bézout is checked, the lemma of Euclide and the fundamental theorem of arithmetic applies. The equivalents of the prime numbers are the irreducible polynomials , i.e. those which have as same dividers only them or the unit except for a multiplicative constant. The decomposition in irreducible polynomials is the possible factorization most complete.

The equivalent of arithmetic modular is focused on the rings quotientés by ideals first (i.e. ideals generated by irreducible polynomials). As previously these ideals have a structure of body. The quotients are called Corps of rupture because are more small body containing a root polynomial. This approach, making it possible to define a finished Extension of the body K defines the basic tool of the Théorie of Welshman.

An example of application is the following: the cyclotomic polynomials correspond to the decomposition in irreducible factors of the polynomial of the roots of the unit $X^n-1$. The analysis of these polynomials makes it possible to determine all the Polygone S constructible with the rule and the compass.

Entireties of Gauss

See also: Whole of Gauss

The entireties of Gauss noted Z correspond to the numbers of the form U + I. v or U and v is selected entireties. They form an Euclidean ring, the definition is given by the following proposal, if NR ( X ) indicates the algebraic standard i.e. the sum of the square of the whole part and the square of the imaginary part:

The application which with an entirety associates its algebraic standard is well an application of the entireties of Gauss in an ordered unit, namely that of the positive whole . This standard corresponds graphically to the square of the distance between the origin and the entirety of Gauss.

To say that Euclidean division exists means that there exists an entirety of Gauss at a distance lower than 1 of the Complex number has / B . The figure attached famous by a red bottom the square of top of the entireties of Gauss and container has / B . The figure shows that there exists always at least an entirety at a distance lower than 1 of has / B . In the illustrated case, there are three checking this property. The unicity of the solution is not a requirement with the existence of an Euclidean division.

Once again, Euclidean division brings an arithmetic analog to the two preceding cases.

The applications are numerous. Dedekind, for example, found a elegant proof of the Théorème of the two squares of Fermat starting from this unit. In rule general, the quadratic equations diophantiennes are solved well in this unit. Gauss used this arithmetic to show the quadratic Loi of reciprocity.

In general, a whole of this nature, called ring of whole algebraic, does not have Euclidean division. Thus, $\backslash \; mathbb\; Z$ is not Euclidean.

Other Euclidean rings

One knows all the quadratic whole rings of Euclidean for the standard induced by the standard of $\backslash \; mathbb\; \{C\}$. On the other hand, an open-ended question is to know if there exists an infinity of algebraic rings of Euclidean entireties.

• If K is a commutative body, K X the ring of its formal series is also Euclidean for the valuation: v (P) = smaller degree of X in P.
• If has is an Euclidean ring and if S is part of has stable for the multiplication. The localization of has compared to S is also an Euclidean ring.

Definitions

Either has a unit ring Commutatif and integrates.

* an Euclidean stathme is an application v of has - {0} in the whole of the positive entireties NR , such as if has and B is two elements of has such as B divides has , then v (b) $\backslash \; scriptstyle\; \{\backslash \; Leq\}$ v (A).

This application is used as measurement for Euclidean division.

* an Euclidean stathme v defines a Euclidean division if, and only if:

* a ring is a Euclidean ring if, and only if there exists an Euclidean stathme defining an Euclidean division.

The property of the stathme is not always regarded as necessary for the definition of an Euclidean ring.

First properties

In the continuation of the article has is a unit and just commutative ring, v a valuation defining an Euclidean division.

* an Euclidean ring is always principal.

More precisely, any element of minimal valuation among the elements of the ideal is a generator of the ideal. It is a direct consequence of the property of Euclidean division.

The three following properties are valid for any principal ring, therefore in particular for an Euclidean ring:

* a principal ring checks the Identité of Bézout: if has and B are two elements of has not having other common dividers that the elements of the Groupe of the units of the ring, then there exists U and v elements of the ring such as has . U + B . v = 1.

* If has is first, then has /a. has is a body.

* a principal ring checks the Lemme of Euclide: are has , B and C three elements of has such as has divides B . C and such as there do not exist other dividers commun run with has and with B only elements of the group of the units. Then has is a divider of C .

Arithmetic

Euclidean Stathme

For each of the three examples developed in the article, there exists an application of measurement remainder associated with Euclidean division. They are respectively the absolute value, the degree of the polynomial and the algebraic standard. These applications v check all the three following properties:

they are with values in the positive entireties

if has and B is nonnull elements of the ring, then v ( has . B ) = v ( has ) + v ( B )

v (A) is equal to zero if, and only if, has is an element of the group of the units of the ring.

Such an application is called a Valuation ring. To obtain an Euclidean ring, this condition is sufficient, but it is not necessary. The properties of the stathme make it possible to show the existence of a decomposition in factor first.

To include/understand the need for such a property, it is enough to consider the ring has Z / P or P = 2. X - 1. This ring is the ring generated by 1/2, it is not nevertheless equal to the body of the dyadic numbers because no completion is carried out here. Then 1/2 is not decomposable in prime numbers .

It is possible to notice that if v is the stathme of an Euclidean ring then:

* Is has and B two elements of has then if v ( has . B ) = v ( has ), then B is an element of the group of the units.

It is always possible to standardize the stathme by the translation v' ( X ) = v ( X ) - v (1) + 1 where X is an element not no one of has , in this case:

* an element not no one U of has is element of the group of the unit if, and only if, v ( U ) = 1. Moreover, 1 is the value minimal which takes v .

Definitions

It is then time to define the usual concepts in arithmetic Euclidean rings:

* an element p of has is called first if, and only if, the ideal p . has is first, i.e. it is different from has and if X . is element of p there. has , then either X or is element of p there. has . Here X and indicates two elements unspecified of there has .

* Is has and B two elements of has then a lowest common multiple of has and of B is a generator of the ideal intersection of has . has and of B . has .

* Is has and B two elements of has then a highest common factor of has and of B is a generator of the ideal generated by has and B .

The definitions chosen here have the advantage of existing in all the rings. The fact of being in an Euclidean ring adds certain properties. For example as the ring is principal, very ideal first is maximum, and p is first if and only if he does not admit other dividers only itself and 1, with the group of the unit near. the existence and unicity with the group of the unit close to the PC and pgcd are also guaranteed.

In the case of the principal rings, an element is first if and only if it is irreducible. An element p is known as irreducible if and only if some is has and B of has p = has . B implies that either has or B or element of the group of the units.

To go further, it is necessary to establish the fundamental theorem of the arithmetic one.

Fundamental theorem of the arithmetic one

See also: fundamental Theorem of arithmetic the

* Is has an Euclidean ring, then each element of has can be written like a product of elements first with the invertible elements close to a single way.

The expression with the invertible elements close means that the subsitution of irreducible factor by another irreducible factor different only by the product from an element of the group of the unit is not regarded as a different decomposition.

Thus, for all has has not no one, there exists an invertible element U and p _i elements first such as:

$a\; =u\; \backslash \; prod\_\; \{i=1\}\; ^n\; p\_i^\; \{v\_\; \{p\_i\}\; (A)\}$ Where $\backslash \; scriptstyle\; \{v\_\; \{p\_i\}\}$ indicates the application which with an element has associates the power of the element first p _i which appears in the decomposition in elements first of has . This application is called the p-adic valuation of p _i.

In the three examples of the article, the elements first are: prime numbers, polynomials irreducible or the prime numbers of Gauss.

To show it, it is enough to notice that the ring is principal and that any principal ring is factorial (cf principal Anneau). Note:: If the ring has of a stathme and an Euclidean division built starting from the stathme, it is possible to adapt the demonstration of the fundamental theorem of arithmetic for the general case of the Euclidean rings.

Consequences

* an Euclidean ring is noethérien :

Consequently any increasing chain of ideals is stationary starting from a certain row. Another manner of describing cettte property amounts saying that any nonempty family of ideals has a maximum element for inclusion.

* Any Euclidean ring is factorial Anneau :

Consequently, there exists a decomposition in factors first single (with the group of the units near). the elements first and irreducible are confused, the pgcd and PC exist always east are single (with the group of the units near). The lemma of Euclide as well as the identity of Bézout are checked.

* an Euclidean ring is principal :

To determine a pgcd of two elements, one can use a Algorithme of Euclide: if has = bq + R , a pgcd of ( B, R ) is also a pgcd of ( has, B ). v (R) being lower v (b) , one is assured that the method has an end: the remainder ends up being null.

See too

Random links:

Roland (company) | White pine of Korea | Borek (Wrocław) | Alain Baudry | Maurice Dekobra | Champions_(jeu_de_role-playing)

© 2007-2008 speedlook.com; article text available under the terms of GFDL, from fr.wikipedia.org