Normal extension

In Mathematical, a normal extension L of K is a particular case of Extension of body. An extension is known as normal if and only if all Morphisme of body leaving invariant K is a Automorphisme of L .

This property is used to define a Extension of Welshman. By definition, in the case of a separable Extension then any normal finished extension is a Welshman extension.

Motivation

The fundamental Théorème of the Welshman theory watch who there exists a fertile correspondence between a Extension finished L on K and its Groupe of Welshman, if the group is sufficiently rich . The group of Welshman indicates the whole of the Automorphisme S of body of L leaving K invariant.

That is to say P a polynomial with coefficients in K with a root R in L . Each morphism of body of L has as an image of R another root of P. So that the group of Welshman is sufficiently rich , it is necessary that all these roots are in L . What results in the fact that any morphism has as an image L .

An other condition is necessary, it is related to the separability and is treated in the separable article Extension. If the two conditions are met, then the extension is known as of Welsh and the conditions of the fundamental theorem are met.

In the case or separability is guaranteed, for example because the body K is perfect, then it is possible to find a good normal extension. For example, in the case of a polynomial with coefficients in a body K perfect, there exists a smaller normal extension containing the roots of the polynomial, it is the Corps of decomposition polynomial.

The theorem fondatemental of the Welshman theory has many applications. Let us quote for example the Théorème of Abel who gives a requirement and sufficient so that a polynomial equation is resolvable by radicals.

The concept of normality is developed in Welshman the article Extension.

Definition

That is to say K a body, L an extension of body of K and Ω the algebraic Fence of K . The extension L is known as normal if and only if any morphism of body of L leaving invariant K and with value in Ω is an automorphism of L .

Properties

The properties of the normal extensions are developed and shown in the article Extension of Welshman. Let us point out the principal properties here. Here L indicates a finished Extension of K .

* the cardinal of the Groupe of Welshman is lower or equal to the dimension of L on K .

I.e. the whole of the automorphisms of L leaving K invariant and whose image of L is equal to L is always of cardinal lower or equal to the dimension of L .

* One supposes that L is a finished extension (i.e. the dimension of L on K is finished). The fact that any irreducible polynomial with coefficients in K having at least a root in L has all its roots in L is a requirement and sufficient so that the extention L is normal on K .

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