Muertos malvados II

In Mathematical and more precisely in Algebra, the theory of Welshman is the study of the extensions of body Commutatif S, by the means of a correspondence with groups of transformations on these extensions, the groups of Welshman. This fertile method, which constitutes the historical example, essaimé in good of other branches of mathematics, with for example the differential Théorie of Welshman, or the Théorie of Welshman of the coatings.

This theory was born from the study by Welsh Évariste of the algebraic equations. The analysis of Permutation S of the roots makes it possible to clarify a requirement and sufficient resolvability by radicals. This result is known under the name of Théorème of Abel-Ruffini.

The essential tools of the theory are the extensions of body and the groups of Welshman.

The applications are very varied. They extend from the resolution of old conjectures like the determination of the constructible Polygone S with the rule and the compass shown by the Théorème of Gauss-Wantzel to the algebraic Géométrie through, for example, the Théorème of the zeros of Hilbert.

History

Genesis

The Welshman theory sees his origins in the study of the algebraic equations. It is reduced to the analysis polynomial equations. An approach by changes of variables and substitutions allowed mathematicians like Al-Khwarizmi (783 850) , Tartaglia (1499 1557) , Cardano (1501 1576) or Ferrari (1522 1565) to solve all the cases until the degree four. This approach does not make it possible to further go and two centuries will be necessary to bring new ideas.

Gauss and cyclotomic polynomials

detailed Paragraph: History of the cyclotomic polynomials

Gauss (1777 1855) uses the cyclotomic polynomials to contribute a share to an open problem since antiquity: that of the Construction to the rule and the compass of regular Polygon S. It builds in particular the heptadécagone, polygon regular with 17 with dimensions. Its approach, typically galoisienne well before the discovery of the theory, is worth to him the nickname of prince of the mathematicians .

Its work is supplemented by Wantzel (1814 1848) , which gives a requirement and sufficient constructibility of the regular polygons and shows the impossibility of the Trisection of the angle and the Duplication of the cube.

Theorem of Abel-Ruffini

detailed Paragraph: History of the theorem of Abel-Ruffini

In the general case, the quintic equation does not admit a solution by radicals. This is why a step using substitutions and changes of variables becomes sterile. Lagrange (1736 1813) and Vandermonde (1735 1796) use the concept of permutation at the end of the 18th century and have a presentiment of the importance of this tool within the framework of the polynomial equation.

Ruffini (1765 1822) is the first to envisage the impossibility of the general solution and that the comprehension of the phenomenon lies in the study of the permutations of the roots. Its demonstration remains nevertheless not very rigorous and partial. The Norwegian mathematician Abel (1802 1829) publishes a demonstration in 1824 which ends up convincing the scientific community. She does not propose at the time of requirement and sufficient of resolvability.

Welsh Evarist

By studying the problem of the algebraic equation, Welsh (1811 1832) highlights the first elements of the theory which bears its name now. Its writings are lost or fall into the lapse of memory. A report is finally found by Liouville (1809 1882) which presents it to the Academy of Science in 1843. Welshman work reaches in extremis the posterity then.

Welshman, for the first time in the history of mathematics, highlights an abstract structure that it calls group. With the difference of its predecessors, he does not study a particular incarnation like the permutations of Lagrange or the cyclic groups of Gauss, but a general structure defined by a unit and a law.

This step, particularly innovative, is at the origin of the modern algebra. Liouville speaks about it in the following terms: This method, really worthy of the attention of the geometricians, would only be enough to ensure our compatriot a row in the small number of the scientists who deserved the title of inventor.

Algebraic structures

The major Welshman contribution, i.e. the use of an algebraic structure like fundamental tool, is quickly included/understood by the mathematical community. Cauchy (1789 1855) publishes twenty-five articles on the groups including one on its famous theorem. Cayley (1821 1895) gives the first abstract definition of a group. Lastly, Jordan (1838 1922) diffuses the Welshman ideas widely. Its book of 1870 presents Welshman work like a general theory on groups, whose theorem on the resolution of the equations is only one application. In France, the Welshman theory is identified with that of the groups at that time.

Other structures are highlighted, particularly in Germany. Independently of Welshman work, Kummer (1810 1893) studies rings and discovers the ancestor of the concept of Idéal. Kronecker (1823 1891) and Dedekind (1831 1916) develop the premises of the theory of the rings and the body. Kronecker establishes the bridge between the schools German Frenchwoman. It gives the modern definition of group of Welshman starting from Automorphisme S of body.

At the end of the 19th century Weber (1842 1913) carries out a synthesis of various work. The Welshman theory is then for the first time identified with that of the bodies Commutatif S.

Welshman theories

A new axis of analysis enriches the theory by Welshman. In 1872 Klein (1849 1925) is fixed like objective to classify different the Géométrie S from the time. It releases, in its famous Programme of Erlangen, the general principle that a geometry is defined by a space and an operative group on this space, called group of the Isométrie S. a bridge is thus established between the theory of the groups and the geometry. These first groups correspond to groups of Dregs and do not belong directly to those of the Welshman theory.

In 1877 Klein notices that the group of the isométries leaving invariant the Icosaèdre is isomorphous with the group of Welshman of a quintic equation. The Welshman theory extends to the algebraic Géométrie. The groups of Welshman take then the form of coatings so called coating of Welshman. David Hilbert (1862 1943) studies the quadratic bodies of numbers and contributes a major share to the theory by showing its famous theorem of the zero. This theorem has also a geometrical interpretation on the algebraic varieties. The theory is now enriched by a new branch: the geometrical Welshman theory. It proves particularly fertile.

Work of Hilbert opens other branches of the Welshman theory. The theorem of the zeros allows the study of the first groups of Welshman of an infinite nature. Its theorem of irreducibility opens the opposite problems . It is stated in the following way: if G is a group then is the group of Welshman of an extension?

Finally work of Picardy (1856 1941) and Vessiot (1865 1952) opens another way for the study of the groups of Welshman of an infinite nature, the differential Théorie of Welshman.

Contributions of the 20th century

Work of Hilbert opened the study of the cases where the group of Welshman is of an infinite and commutative nature. This vast subject takes the name of Théorie of the bodies of classes. It is now completed and is often regarded as one of more nice successes of mathematics of the century.

The final formalization of the Welshman theory is given by Artin (1898 1962) . The addition of the Linear algebra allows a clearer and concise exposure. The theory now uses all the great structures of the algebra, the groups, the rings, the bodies and the vector spaces. It now has important ramifications in algebraic geometry.

It is the base of a major quantity of the great mathematical achievements of the 20th century. The alliance of the geometry and the algebra is almost systematically used. One can quote for example work of the mathematicians Jean-Pierre Serre Médaille fields 1954 and Grothendieck Médaille fields 1966 with a recasting of the algebraic geometry, Faltings Médaille fields 1986 for this work on the modules of Welshman showing the theorem of Mordell or Laurent Lafforgue Médaille fields 2002 on the Programme of Langlands, a generalization of the theory of the bodies of classes.

Examples

Small theorem of Fermat

See also: Algebraic theory of the numbers

The small theorem of Fermat indicates to us that if has is an entirety and p a prime number then:

$a^p\; \backslash \; equiv\; has\; \backslash \; (\backslash \; bmod\; \backslash \; p)$ It is possible to show this theorem by noticing that F _p the quotient of the whole of the integers by its Idéal generated by p is a body, because p is a prime number. ( F _p^*.) is a finished group of cardinal p -1. The theorem of Lagrange ensures that any element of this group to the power p -1 is equal to the unit, which shows the theorem.

Conclusion: This case is particularly easy because the structure of the body is simple. It illustrates nevertheless the fact that a structure of body is a useful tool in Algebraic theory of the numbers. Other theorems of Arithmetic modular as the quadratic Loi of reciprocity require a comprehension much major of the structure of the bodies. This is why the demonstration could not be found in spite of their efforts by Euler (1707 1783) or Lagrange and until had to be waited cyclotomic Gauss and its polynomials to conclude.

Duplication of the cube

See also: quadratic Extension, quadratic Turn of extension

Either L the whole of the elements of the form has + B . √2 where has and B is rational.

Montrons that L is a body: L is clearly stable for the addition and the passage on the other hand, and is thus an additive group. It is stable for the multiplication and the passage contrary to the nonnull elements, indeed:

$\backslash \; forall\; has,\; a\text{'},\; B,\; b\text{'}\; \backslash \; in\; \backslash \; mathbb\; \{Z\}\; \backslash \; quad\; (a+b\; \backslash \; sqrt\; \{2\}).\; (a\text{'}+b\text{'}\; \backslash \; sqrt\; \{2\})\; =aa\text{'}+2bb\text{'}+\; (ab\text{'}+a\text{'}\; b)\; \backslash \; sqrt\; \{2\}$ $et\; \backslash \; quad\; \backslash \; forall\; has,\; B\; \backslash \; in\; \backslash \; mathbb\; \{Z\}\; \backslash \; quad\; if\; \backslash ;\; \backslash \; neq\; 0\; \backslash \; has;\; or\; \backslash ;\; B\; \backslash \; neq\; 0\; \backslash \; quad\; then\; \backslash \; quad\; (a+b\; \backslash \; sqrt\; \{2\}).\; \backslash \; frac\; \{a-b\; \backslash \; sqrt\; \{2\}\}\; \{a^2-2b^2\}\; =1\; \backslash \; quad\; because\; \backslash \; quad\; a^2-2b^2\; \backslash \; neq\; 0$ These two last proposals show that L is a subfield of the real numbers. L is also a vector Space of dimension two on the rational numbers, because it has for bases 1 and √2. Such a body is called a quadratic Extension.

Montrons that L does not contain the cubic root of two: Is L an element of L , then of L ² is a linear Combinaison with rational coefficients of 1 and √2 because these two elements form a base and L ² is an element of L . there thus exist two rational $\backslash \; alpha$ and $\backslash \; beta$ such as the following equality is true:

$l^2-\; \backslash \; alpha\; L\; -\; \backslash \; beta\; =\; 0\; \backslash ;$ And for any element L of L there exists a polynomial of degree lower or equal to two having for root L . However, the smallest degree of the polynomial not no one with rational coefficients which cancels the cubic root of two is three, and the proposal is shown.

Conclusion: Wantzel showed that the constructible numbers with the rule and the compass are either in a quadratic extension, or in a quadratic extension whose coefficients are taken in a quadratic extension and so on. One then speaks about quadratic Tour of extension. It is possible to show by a reasoning similar to that presented here that the cubic root of two is not element of a tower of quadratic extension. For this reason the duplication of the cube is impossible. The wise choice of particular bodies is the key of the resolution of this ancient conjecture.

Cubic equation

See also: Theorem of Abel-Ruffini

Let us consider an example of cubic equation:

$P=0\; \backslash \; quad\; with\; \backslash \; quad\; P=X^3-3X+1\; \backslash ;$ Determination of an element of the group of Welshman: the polynomial P is an irreducible polynomial with Coefficient S rational. The Welshman theory indicates to us that there exists a body L which is a extension the rational ones containing all the roots of P . This extension is of dimension six. Moreover, there exists a subfield K of L such as L is of dimension three on K and J the cubic first root of the unit is element of K . The group of Welshman of L on K is the whole of the Automorphisme S of body of L leaving invariant any element of K . The Welshman theory indicates to us that this group has three elements, that is to say G an element of the group different from the identity.

Diagonalisation of G : the theorem of Lagrange ensures us that G ³ is equal to the identity. If L is an element not no one of L , then L , G (L) and g² (L) forms a base of L on K . Considering G as linear operator, his characteristic Polynôme is X³ - 1 and its eigenvalues are 1, J and j² . There exists a base ( U , v , W ) of L on K consisted of clean vectors, because the number of eigenvalues is equal to the dimension of L on K . There is thus G ( U ) =j· U , G ( v ) =j²· v and G ( W ) = W . Moreover, let us note that 1 + J + j² = 0.

Determination of the image of the roots by G : Is X ₁, a root of P . The image of a root per G is a root, indeed:

$P\; \backslash \; left\; (G\; (x\_1)\; \backslash \; right)\; =g\; (x\_1)\; ^3-3g\; (x\_1)\; +1=g\; (x\_1^3-3x\_1+1)\; =g\; (P\; (x\_1))=g\; (0)\; =0$ One from of deduced that X ₁, G ( X ₁) and G ² ( X ₁) is the three roots of P . One can repésenter X ₁ like summons clean vectors of G : X ₁ = U + v + W (light abuse notation: W will be in fact 0, not forming one base like above more). The three following equalities are then checked: $x\_1=u+v+w\; \backslash \; quad,\; \backslash \; quad\; G\; (x\_1)\; =j\; \backslash \; cdot\; u+j^2\; \backslash \; cdot\; v+w\; \backslash \; quad\; and\; \backslash \; quad\; g^2\; (x\_1)\; =j^2\; \backslash \; cdot\; u+j\; \backslash \; cdot\; v+w$

Calculation of the values of the roots: It is enough to use the Relations between coefficients and roots to show that:

$x\_1+g\; (x\_1)\; +g^2\; (x\_1)\; =\; (1+j+j^2)\; u+\; (1+j+j^2)\; v+3w=0\; \backslash \; quad$, therefore $\backslash \; quad\; w=0\; \backslash ;$

$x\_1\; \backslash \; cdot\; G\; (x\_1)\; +g\; (x\_1)\; \backslash \; cdot\; g^2\; (x\_1)\; +g^2\; (x\_1)\; \backslash \; cdot\; x\_1=\; (u+v)\; (ju+j^2v)\; +\; (ju+j^2v)\; (j^2u+jv)\; +\; (j^2u+jv)\; (u+v)\; =-uv=-3\; \backslash \; quad$, therefore $\backslash \; quad\; u^3v^3=1\; \backslash ;$

$x\_1\; \backslash \; cdot\; G\; (x\_1)\; \backslash \; cdot\; g^2\; (x\_1)\; =\; (u+v)\; (ju+j^2v)\; (j^2u+jv)\; =-1\; \backslash \; quad$, therefore $\backslash \; quad\; u^3+v^3=-1\; \backslash ;$

One from of deduced that U ³ and v ³ checks the equation X ² + X + 1 = 0. What makes it possible to conclude that X ₁ is equal to 2 cos (2$\backslash \; pi$/9), 2 cos (8$\backslash \; pi$/9) or 2 cos (14$\backslash \; pi$/9).

Conclusion: the group of Welshman, allows the resolution of the cubic equation by a diagonalisation of a endomorphism. The method is generalizable if and only if the group of Welshman has good properties, makes of it if it is resolvable.

Synthesis

These examples have a common point, they are the properties of the algebraic structures which make it possible to find the solutions. For the first example, the property shown by Lagrange on the groups (and thus multiplicative groups of the bodies) finished make it possible to conclude. In the second example, in fact the properties associated on dimension with vector space are used. In the third case, the properties of the bodies and their extensions are used, groups with the theorem of Lagrange and that of the vector spaces with the properties of Réduction of endomorphism if the minimal polynomial is divided.

The Welshman theory offers a richness in the algebraic structures making it possible to solve many very different cases and in distant fields.

Applications

Algebraic theory of the numbers

The algebraic theory of the numbers is the study of the roots numbers of a polynomial with whole coefficients, called algebraic numbers.

The Welshman theory is essential here because it offers the most adequate structure of analysis, namely the finished Extension smallest containing the studied numbers. A subset plays a particular part: that of the algebraic whole , they corresponds to the generalization of the entireties in the extension. The study of this unit adds to the Welshman theory of many properties resulting from the theory of the rings. The algebraic entireties play a big role for the solution of equations of modular Arithmétique or diophantiennes.

One can quote like application of the Welshman theory to this field, the Théorème of Gauss-Wantzel which determines all the constructible regular polygons with the rule and the compass. the Théorie of Kummer applies to the equations diophantiennes and makes it possible to validate the Grand theorem of Fermat for almost all the entireties lower than hundred. Lastly, within the framework of arithmetic modular, the law of reciprocity of Artin generalizes the quadratic Loi of reciprocity of Gauss and résoud the ninth problem of Hilbert.

Cryptography

Cryptography is the discipline which attempts to protect a message. The theoretical framework maintaining more used consists in defining a algorithm which, associated with a key makes it possible to create a new message known as Cryptogramme meaning that it is quantified. The coded message is simple to decipher, i.e. simple to transform into message of origin with a key and difficult without this one for the person who then endeavors to decipher it.

In part of the modern theories of cryptography, the letters of the message are selected in a finished body. The framework is thus that of the Welshman theory.

It is natural that the associated tools are those of the theory. The arithmetic modular one (cf for example the algorithm RSA) is very largely employed. If the simple techniques rest on elementary results like the Théorème of Bézout, the Théorème of the Chinese remainders or the modular Exponentiation, the current developments uses more subtle tools like the elliptic curves (cf an inviolable private key?).

Theory of the algebraic equations

The problems of the theory of the algebraic equations are that which gave rise to the Welshman theory. It supplements the Théorème of Abel-Ruffini by proposing a requirement and sufficient for the existence of an expression by radicals of the roots of a polynomial.

It makes it possible nevertheless to go further. The Théorème of Kronecker-Weber precisely clarifies the structure of the rational extensions associated with the polynomials having roots being expressed by radicals. It then becomes possible to solve explicitly all the equations of this nature.

It has for fields of application all the bodies, offering a powerful tool to the modular Arithmétique. Many laws of reciprocity, of comparable nature that shown per Gauss in the quadratic case are thus demonstrable thanks to the Welshman theory.

Abel then Hermite (1822 1902) worked on another approach: the elliptic functions. They allow, for example, to express the roots of any polynomial equation. The geometrical Welshman theory integrates this concept through the elliptic curved . The Grand theorem of Fermat was shown using methods of this nature.

There exists a a little particular Welshman theory treating differential equations polynomial. This theory takes the name of differential Théorie of Welshman. She studies a particular family of body called differential extension . These bodies have groups of Welshman. The solution of an algebraic equation also corresponds to the analysis of the associated group and allows the solution of a differential equation.

Algebraic geometry

Structures used

Commutative bodies

See also: Body (mathematical), Extension of body

The commutative body is the object of the Welshman theory. It is thus naturally the central structure of the theory.

The most important technique of construction corresponds to the extension, i.e. with a body which contains the body of origin. From the basic body, often smallest, that generated by the unit, which is a cyclic body (built starting from a cyclic Groupe of order a prime number) or that of the rational a new structure is created.

This method allows the creation of a zoology describing the various properties of the structure. A body can thus be for example algebraic, simple, perfect, quadratic, separable, cyclotomic or algebraically closed.

There exist important theorems, like that of the primitive element or that of Wedderburn which ensures that any finished body is commutative.

Vector space

See also: finished Extension

An extension has a structure of vector Space on its basic body. This structure is important for two reasons:

It makes it possible to still classify the study of the various bodies, those of finished size known as finished Extension and the others. Just as in algebra linaire, the first case is infinitely simpler than the other.

It is then a tool which allows the demonstration of many properties by associating the theory the theorems of linear algebras. One can quote for example the Théorème of Gauss-Wantzel whose demonstration is in the paragraph applications of the turns of quadratic extension or the Théorème of Abel-Ruffini who uses a Diagonalisation Endomorphisme.

The case of infinite size is largely more complex, it is partially treated in the Théorie of the bodies of class.

Ring

See also: algebraic Extension

An important tool of the theory is the Polynôme. And the structure of ring is that of the whole of the polynomials. It is used for example to build extensions. An extension is thus often the quotient of the ring of the polynomials by a Idéal generated by an irreducible polynomial.

A polynomial plays a particular part in the theory: the minimal Polynomial which is the unit polynomial of smaller degree which has for root a given element. Thus, an extension is algebraic if all the elements have a minimal polynomial, quadratic if the minimal polynomial of any element is of degree lower or equal to two, separable if no minimal polynomial has multiple root, cyclotomic if the extension is generated by a root of a cyclotomic Polynôme. A body is perfect if any extension is separable.

The algebraic theory of the numbers also often uses subsets of an extension having only one structure of ring, such as for example the algebraic whole .

Group

See also: Group of Welshman, fundamental Theorem of the Welshman theory

This structure is the major contribution of the mathematician bearing the name of the theory.

The group of Welshman is the group of the Automorphisme S of an extension leaving invariant the basic body. Under certain relatively general conditions, the body is entirely characterized by its group of Welshman. An extension satisfying these conditions is known as galoisienne. In particular, if the structure of vector space is of finished size, then the group of an abelian extension is of order the dimension of the group.

As it is largely simpler to study a finished group than a structure of body, the analysis of the group is a powerful method to include/understand the body. The group of Welshman is in the beginning many theorems. One can quote the fundamental theorem of the theory, the theorem of Abel-Ruffini or that of Kronecker-Weber.

Topology

Welshman theories

Classical theory

The traditional term of is largely used, even if it does not have a precise definition. One finds it for example, on the page of presentation of a member of the Academy of Science: Jean-Pierre Ramis. It is also used largely by Daniel Bertrand professor with the university of Paris VI.

It indicates the theory in general recovering the algebraic extensions finished and separable. the theory treats primarily normal extensions and thus galoisiennes. The principal results are the Théorème of the primitive element and the fundamental Théorème of the Welshman theory. This framework allows for example the demonstration of the Théorème of Abel-Ruffini of Gauss-Wantzel or of Kronecker-Weber, it is used in the classification of the Corps finished S.

The extent of this theory covers the state of science at the time of Weber i.e. the end of the 19th century, even if now it is very generally presented with the formalism of Artin. That corresponds a little to the case of the dimension finished for the linear algebra.

Geometrical theory

Opposite theory

It is in general difficult to determine the group of Welshman of a given extension. This remark poses the opposite problems naturally: that is to say is a given group, which the extension on a given body which has this group like groups of Welshman? This question is that which the opposite theory seeks to answer.

In the case of the finished groups, a first result shows that if N is a strictly positive entirety then there exists an extension of the body of rational having for group of Welshman the symmetrical Groupe of order N . For example, the Body of decomposition of the rational polynomial X ⁿ - X - 1 admits for group of Welshman the symmetrical group of order N . The Theorem of Cayley and the fundamental Théorème of the Welshman theory make it possible to deduce from it that, for any group finished G , there exists an extension of a body of numbers (i.e. a finished Extension of the rational numbers) having G for group of Welshman.

The opposite theory seeks to answer three questions:

* G and a body K , does there exist Is a group an extension of K having G for group of Welshman?

* finished G , does there exist Is a group a normal extension of rational having G for group of Welshman?

* finished G and a body K , does there exist Is a group a normal extension of K having G for group of Welshman?

In spite of important progress during the thirty last years of the 20th century, in 2006 the three questions remain very largely open.

Differential theory

Some elementary functions have a nonelementary Primitive. A famous example is the Gaussian defined by the standardized primitive of the function exp (- X ²/2).

It should be noted that whatever the choice of the list defining the elementary functions, there exist always examples where the primitive does not express in algebraic term of compound of elements of the list.

The differential theory of Welshman allows the determination of the whole of the elementary functions whose primitive is still an elementary function. This theory studies particular bodies called differential Corps S. They are the bodies K provided with a derivation $\backslash \; delta$. A derivation is an operation checking the following property:

$\backslash \; forall\; has,\; B\; \backslash \; in\; K\; \backslash \; quad\; \backslash \; delta\; (a+b)\; =\; \backslash \; delta\; (a)+\; \backslash \; delta\; (b)\; \backslash \; quad\; and\; \backslash \; quad\; \backslash \; delta\; (a.b)\; =\; \backslash \; delta\; (a).b\; +\; A.\; \backslash \; delta\; (b)\; \backslash ;$ This milked branch of a family of body, it is thus natural to regard it as a particular case of the Welshman theory. However the analogy further goes and many regards, this theory resembles the classical theory. The principal difference is that, in this context, the group of Welshman is not any more one finished group but in general a algebraic Groupe.

Theory of the bodies of classes

See too

Random links:

Blies-Guersviller | Canton of the Vault-the-Queen | Tollevast | Death before dishonor | Davis Kelvin | Morts_mauvais_II

© 2007-2008 speedlook.com; article text available under the terms of GFDL, from fr.wikipedia.org