Michael Wittmann

In Mathematical, and more specifically in Algebra for the Theory of Welshman, the theorem of the primitive element is one of the basic theorems in the case of the finished extensions of body.

The theorem establishes an equivalence if the extension is finished between the separable concept of Extension, the number of morphisms of the extension in the algebraic Clôture leaving invariant the basic body and the existence of a separable element generating the extension.

This theorem in the case of belongs to the two pillars of the Welshman theory the finished extensions. It establishes an equivalence which introduces the concept of separability. It is used for example for the demonstration of the fundamental Théorème of the Welshman theory.

Motivation

detailed Article: separable Extension

The Welshman theory, has like fundamental structure Welshman the Groupe, this group makes it possible to show a series of geometrical results, like the characterization of the constructible numbers to determine for example the regular polygons constructible with the rule and the compass, the resolution of algebraic equations or to establish results in Théorie of the numbers like the Grand theorem of Fermat in much of particular cases.

Like another example of this group, one can quote the characterization of the finished bodies. The group makes it possible to determine the exact structure of the body as well as the nature of the various polynomials with coefficients in this body.

To allow these demonstrations, the group of Welshman must be sufficiently vast , which means two properties: the extension must be separable and normal. An element of the extension is known as separable if and only if its minimal Polynôme with coefficients in K does not have multiple roots in its algebraic Clôture. An extension is known as separable if all its elements are it. The theorem indicates that it is enough that a quite selected element is separable so that all the extension is it. This property is important because the root of a polynomial is always transformed by an element of the group of Welshman into another root. If there exist multiple roots, then the group becomes too small so that the fundamental Théorème of the Welshman theory can apply.

If the extension is of size finished as a vector Space on the basic body K , if the extension is separable then the number of morphisms of the extension in the algebraic fence is the good. There exists moreover one strong property: there exists an element L of the extension such as L is equal to K ( L ) i.e. L is the smallest extension containing K and L and of course L is separable. One then speaks about simple Extension algebraic. The reciprocal one is also true and is constitutive of the theorem of the primitive element.

Statement of the theorem

That is to say L an extension finished of dimension N on a body K , and Ω the algebraic fence of K . Then the theorem of the primitive element takes the following form:

* the four following conditions are equivalent:

the extension L is separable on K .
the extension is generated by separable elements.
There exists exactly N morphisms of L in Ω leaving invariant K .
L is a simple Extension generated by a separable element.

Note:: There exist other expressions of the theorem of the primitive element, for example: any finished separable extension is simple. Another form is the following one: the extension L is simple if and only if it contains a number finished of extensions of K.

Case of the perfect bodies

Separation is a relatively frequent property in the algebraic extensions. For example if the body K is of characteristic 0 then any extension is separable. The characteristic of a body is null if the reiterated addition of the unit is never null. Consequently, any extension of the body of the rational numbers or the real numbers is separable. All Corps finished has also only separable algebraic extensions.

A body which does not admit that separable extensions is known as perfect.

Demonstration

proposal 1 implies proposal 2

The result is immediate.

proposal 2 implies proposal 3

It is a consequence of the two last proposals shown in the paragraph Morphisme in the algebraic fence.

proposal 3 implies proposal 4

Let us suppose that there exists an element L of L having N images distinct by N morphisms from proposal 3. Then its minimal polynomial is of degree N, and K (L) is a vector space included in L and of the same dimension. Two spaces are equal and the proposal is shown. It is then enough to show the existence of L , which is made at the end of the paragraph.

proposal 4 implies proposal 1

That is to say L a generator of L . L is of order N and thus there exists N morphisms of L in Ω leaving invariant K . That is to say then R an unspecified element of L then L is an extension of K (R) and each morphism of L is the made up one of a morphism of K (R) extended to L and of a morphism of L leaving invariant K (R) . That is to say n_r the number of morphisms of K (R) in Ω leaving invariant K and it many morphisms of L in Ω leaving invariant K (R) . We have the three equalities: n_r.n' = N, n_r is lower or equal to '', is not lower or equal to '' and . '' =n. One concluded from it that n_r is equal to ''. The images of R by the various morphisms of K (R) in Ω leaving invariant K are distinct two to two because if not the morphisms would be confused. The minimal polynomial of R thus admits '' distinct roots. We showed that R is separable.

If K is a finished body, then there exists an element L of L having N images distinct by N morphisms from proposal 3.

If K is a finished body, then the multiplicative group associated with L is a cyclic group. If L is selected among the generating elements of the group, then it has N distinct images by N morphisms. If not, there would exist confused morphisms. And the proposal is shown.

If K is an infinite body, then there exists an element L of L having N images distinct by N morphisms from proposal 3.

Let us consider V_ij the whole of the vectors of L having even image by the IE and it I morphism. V_ij is a vectorial subspace different of L . A property on the unions of the vector spaces watch which the union of V_ij is not equal to L . There thus exists an element L of L which is not element of any V_ij. Its minimal polynomial thus admits N distinct roots. This minimal polynomial has a degree which divides N, according to a property shown in the algebraic article Extension. Its degree is thus exactly N. L is thus generating and separable and the demonstration is finished.

See too

External bonds

Theorem of the primitive element in the mathématiques.net
Theorem of the primitive element in mathreference.com
Theorem of the primitive element in planetmath.org
Theorem of the primitive element in the site of the university of Cornell

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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