Body of decomposition

In Mathematical and more precisely in Algebra in the Theory of Welshman, the body of decomposition of a Polynôme P is smallest Extension of body containing all the roots of P. One shows that such an extension always exists.

A body of decomposition of a polynomial is a finished Extension. If it is separable then it normal is . The body of decomposition is then a Extension of Welshman.

All the Welshman theory applies, such a body profits from powerful theorems, like the Théorème of the primitive element or the fundamental Théorème of the Welshman theory. Many problems are solved then using this structure. One can quote for example the Théorème of Abel or the determination of the Polygone S constructible with the rule and the compass.

Definition

The following notations are used for all the article, that is to say K a body P a polynomial with coefficients in K and Ω the algebraic Clôture of K .

* There exists a smaller extension of body L on K such as the polynomial P is divided (i.e. it is the product of polynomials of the first degree) on L . Minimal means here that any extension F containing all the roots of P is an extension of L . This extension is called body of decomposition of P.

* Is ( l₁ ,…, l_n ) a family F of algebraic elements of Ω. the smallest body of Ω containing the family and K is noted K ( l₁ ,…, l_n ) and is called the body generated by the family F .

L is isomorphous with a subfield of Ω, it is thus possible to identify K with and L with subfields of Ω as proves it the paragraph algebraic Extension and algebraic fence. This identification is carried out in the remainder of the article.

If r₁ ,…, r_n is the roots of P, then L is identified with K ( r₁ ,…, r_n ). The demonstration of the existence of the body of decomposition is in the algebraic paragraph Extension and polynomial.

Note:: There exists another convention, the body of decomposition of a polynomial P on K indicates any extension containing all the roots of P, the minimal body is then called the body of the roots.

Examples

The body of decomposition of the X²+1 polynomial on the body of the real numbers is the body of the complex numbers.

Then let us build the body of decomposition L of the polynomial P = X³ - 2 on the body of the rational numbers. That is to say R the cubic root of two, and J the cubic root of the unit having a positive imaginary component. Then the two other roots are j.r and j².r . No root is rational thus the polynomial is irreducible (indeed any polynomial of degree three which is not irreducible has a rational root).

Let us consider the extension K₁ equal to Q ( R ), i.e. the extension generated by R . As P is irreducible, it is an extension of dimension three isomorph to Q/P (cf the paragraph algebraic Extension and polynomial and whose base is given by (1, R , r² ).

On K₁ the polynomial P has a root R . A division of P by polynomial X - R gives the equality:

P= (X-r) (X^2 + r.X + r^2) = (X-r) (X+ (1/2 + S) .r) (X+ (1/2 - S) .r) \ quad with \; s= \ frac {\ sqrt {3}} {2} I \;

One from of deduced that L is equal to K₁ ( S ) which is an extension of dimension two of K₁ and whose base is given by (1, S ).

One with the equality on dimensions =.= 3 X 2 = 6 (cf Definitions and first properties of the algebraic extensions). One from of deduced a base from L (1, R , r² , S , s.r , s.r² ).

Note:: The method presented here is generic, it can be used to build bodies of decompositions.

Properties

a body of decomposition is finished

This property is shown in the algebraic paragraph Extension and on-body.

Is two bodies of decomposition of P on a body K , then they are isomorphous.

This property is shown in the algebraic paragraph Extension and on-body.

If a body of decomposition is generated by separable elements, then it is separable and simple.

The body of decomposition is finished, if it is generated by separable elements then the Théorème of the primitive element applies. Consequently, the extension is separable and simple.

If the body of decomposition is separable and P irreducible, then it is galoisien.

That is to say ( r₁ ,…, r_n ) the roots of P. Alors L is equal to K ( r₁ ,…, r_n ). Any morphism of L in Ω permutes the roots, therefore leaves stable L what shows that the extension is normal. L is an extension separable and normal, it is thus of Welshman.

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
Welshman correspondences in les-mathématiques.net

References

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