Simple extension

In Mathématiques and more precisely in Algèbre in the case of the Théorie of Welshman, a Extension of body L of a body K is known as simple if and only if there exists an element L of L such as L is equal to K .

A simple extension is finished if and only if L is algebraic. It is even the definition of the algebraic character of an element.

The Théorème of the primitive element watch which a finished extension is simple if and only if the extension is separable.

Motivation

Two reasons return the concept of extension simple interesting:

The simple extensions are a particular case of extensions of body which can be the subject of a complete classification. Either the generator of the extension is transcendent (i.e. the extension is not finished) and the extension is isomorphous with the body of the rational fractions, or the generator is algebraic (i.e. the extension is finished) then the extension is isomorphous with a quotient of the ring of the polynomials with coefficients in the basic body by a Idéal generated by a irreducible Polynôme.

The Théorème of the primitive element gives sufficient conditions so that an extension is simple. If the extension is finished and separable, i.e. generated by elements having minimal polynomials without multiple roots, then the extension is always simple. However the case of a separable extension is common; for example if the basic body is of characteristic null or if it is finished then the extension is always separable.

Definition

That is to say L an extension of body of K and Ω the algebraic Fence of K . These notations are used in all the article.

* the extension L is known as simple if and only if there exists L an element of Ω and a Automorphisme of body between L and K (L) leaving invariant K .

* Is L a simple extension and G an element of L such as L is equal to K (G) . Then G is called generating L on K .

As shown in the algebraic article Extension, it is then possible to identify K and its image in the agebric fence and L with K (L) . This identification is subsequently carried out article.

Examples

the body of the complex numbers is a simple extension of dimension two of the real numbers. It is generated by imaginary pure the I .

The demonstration is given in the article Extension of Welshman.

the body generated by the cubic root of two and imaginary pure the I is a simple extension of the body of the rational numbers.

Indeed, a demonstration is given in the article Extension of Welshman.

It is possible to realize by a more direct approach it. The body of the rational numbers is a perfect Corps (separable cf Extension) i.e. no minimal polynomial admits multiple root. The Theorem of the primitive element watch whereas any finished algebraic extension is simple. The extension is finished because it is an extension of two algebraic elements.

It is still possible to realize by a more calculative method it. The body is indeed the extension of the algebraic number R cubic nap of root of two and of imaginary pure the I . It is enough to notice that R is root of a polynomial with rational coefficients of degree six. The algebraic article Extension watch whereas the simple extension of R is of dimension six with for base (1, r, r²,…, r⁵) if the extension is regarded as a vector space on the body of the rational numbers. It is then enough to check that I and the cubic root of two is linear combinations of this base. If the method is calculative, it makes it possible nevertheless to find the result without powerful theorem.

the body of the real numbers is not a simple extension of the body of the rational numbers.

The extension contains at least a transcendent number, for example

\ pi

and an algebraic number of order two, for example the square root of two.

More generally:

All quadratic Extension is simple.

This property comes from the definition of a quadratic extension.

All finished Extension separable is a simple extension, and thus any extension finished on a perfect Corps is simple.

It is a direct consequence of the theorem of the primitive element. In corollary, one with the following example:

All separable Corps of decomposition is a simple extension.

Properties

* If the extension separable and is finished, then the extension is simple and there exists exactly N Morphisme S of body of L in Ω leaving invariant K .

* If the extension is simple and if there exists a separable generator, then the extension is separable.

* If there exists exactly N morphisms of body of L in Ω leaving invariant K . Then the extension is simple and separable.

They are three immediate consequences of the Théorème of the primitive element.

*Si the extension is simple and not finished, then the extension is isomorphous with the rational bodies of the fractions on K .

Indeed, is G a generator of L , it exists a single manner of prolonging the application of L in K (X) the body of the rational fractions which with G associates X in a morphism of body. It is easy to check that it is an isomorphism.

Let us suppose the extension simple and finished, G a generator of L and P is then the minimal polynomial of G to coefficients in K . This polynomial exists according to the paragraph Définitions and first properties of the algebraic extensions.

* L is then isomorphous with the quotient of the ring of the polynomials K '' by the Idéal generated by P, the minimal polynomial of a generator.
It is a direct consequence of the demonstration of the first proposal of the algebraic paragraph Extension and polynomial.

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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