Body of rupture

In Mathematical and more precisely in Algèbre within the framework of the Théorie of Welshman a body of rupture of a Polynôme with coefficients in a body K is a algebraic Extension of K containing at least a root of the polynomial.

The bodies of ruptures are not in general those used in the context of the Welshman theory, because they do not have the good properties necessary to apply the fundamental Théorème of the Welshman theory. On the other hand they are a stage in the construction of the Corps of decomposition S, which them if the criterion of separability is assured make it possible to apply the fundamental theorem.

Motivation

That is to say K a body and P a polynomial with coefficient in K . does a natural question arises, exist an algebraic extension L of the body K containing one or more roots of the polynomial? The answer is positive, moreover it is possible to build a finished extension having these properties. This extension is called a body of rupture .

The technique used consists with quotienter the ring of the polynomials by a Idéal the main thing and more precisely by a maximum ideal, i.e. an ideal contained in any other ideal that the ring K ''. This technique makes it possible to build a new body. It is then necessary to identify the body K with a subfield of L to make it possible the polynomial to operate on L . That is to say ( L , J ) an extension of body of a body K and P a polynomial to coefficients in K . If one notes K^* the image of K by J , then there exists a natural isomorphism $\ phi$ of K '' in K^* '' defined by:

Soit \; (a_i) \ in K^n \; such \, that \; P= \ sum_ {I \ in} a_i x^n \; then \; \ varphi (P) = \ sum_ {I \ in} J (a_i) x^n

The identification of K and K^* makes it possible to identify K '' and K^* '' through the isomorphism of ring φ. P appears then as an element of L '' to coefficients in K included in L. a polynomial of K '' thus operates naturally on L . It is then possible to speak about root of P in L .

If a body of rupture has the merit to exist, on the other hand it does not contain, in general the entirety of the roots of P . It then appears necessary to reiterate the operation until an algebraic extension containing all the roots is built. One speaks then about Corps of decomposition.

Definition

That is to say K a body and P a polynomial with coefficients in K . A body of rupture of P is an extension L of K such as the polynomial admits at least a root in the extension.

Attention much of people uses the following definition unconsciously:

one of the smallest body containing a root.

And they often tend to call it the body of rupture (whereas even with this definition they can be several), therefore to be wary, certain results can appear false whereas does it of it is the latter who trails in the head of the author (confers this article, you will see).

Examples

Maybe in the body of the real numbers, the polynomial X ² + 1. In its body of coefficients, the polynomial does not contain any root. Indeed, any square of the body of the real numbers is strictly positive. One (and not it) natural body of rupture of this polynomial is that of the complex numbers.

Maybe in the body of the rational numbers the polynomial X ³ - 2. Extension of body generated by R the cubic root of two. It is an extension which appears as a vector Space of dimension three having as bases (1, R , R ²). However this extension indeed does not contain all the roots of the polynomial, there are two having a complex component and which are not element of the body.

Properties

Is P an irreducible polynomial of degree N on K , then it exists a body of rupture L . Its dimension is equal to the degree of the polynomial.

Is P a polynomial on K , then it exists a body of rupture L on K .

Either L the body of rupture of an irreducible polynomial P of degree N , then the first proposal ensures us that there exists a body of rupture. There exist also subfields of the algebraic Clôture Ω containing a root of P . The algebraic fence of a body is a on-body of K such as all the polynomials with coefficients in on-body is divided, i.e. break up into product of polynomials of the first degree. If $\ alpha$ is a root of P in Ω then K is also a on-body of K containing a root of the polynomial. The following proposal establishes the bond between the body of rupture and the subfields of the isomorphous algebraic fence to the body of rupture.

There exists with more N morphisms of L in Ω. If P is a separable polynomial , then there exists exactly N morphisms.

A polynomial is known as separable if he does not admit a multiple root. It is in often the case for an irreducible polynomial, it is always true for the bodies of the rational numbers of the real numbers or for the finished bodies. The last property is shown in the article on the separable extensions.

See too

External bonds

a short presentation of the algebraic extensions by Bernard Stum University of Rennes 1 2001
a course of DEA on the Welshman theory per Alain Kraus University of Paris VI 1998
Body of the roots in les-mathématiques.net

References

R. and A. Douady Algebra and theories galoisiennes Cedic/Fernand Nathan 1978

S. Lang Algebra Dunod 2004

P. Samuel Algebraic theory of the numbers Hermann Paris 1971

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