[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/all2en\/wiki32\/algebra-simple-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/all2en\/wiki32\/algebra-simple-wikipedia\/","headline":"algebra simple \u2014 wikipedia","name":"algebra simple \u2014 wikipedia","description":"before-content-x4 In mathematics, a algebra (associative unit) on a commutative body is said simple If its underlying ring is simple,","datePublished":"2019-03-01","dateModified":"2019-03-01","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/all2en\/wiki32\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/all2en\/wiki32\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/44a4cee54c4c053e967fe3e7d054edd4?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/44a4cee54c4c053e967fe3e7d054edd4?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":100,"height":100},"url":"https:\/\/wiki.edu.vn\/all2en\/wiki32\/algebra-simple-wikipedia\/","wordCount":3401,"articleBody":"     (adsbygoogle = window.adsbygoogle || []).push({});before-content-x4In mathematics, a algebra (associative unit) on a commutative body is said simple If its underlying ring is simple, that is to say if it does not admit a bilateral ideal other than {0} and itself, and if it is not reduced to 0. Whether A is a simple ring, so its center is a commutative body K , and considering A like an algebra on K , SO A is a simple algebra on K .        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4Thereafter, we designate by k a commutative body, and any algebra on K is supposed to be of finished dimension on K    An algebra on K is said central If it is not reduced to 0 and if its center is its subanary K .1.Either A simple algebra on K . Then the center of A is an overly (commutative) WITH of K , A can be considered as an algebra on WITH , And A is a simple central algebra (in) on WITH . Thus, part of the study of simple algebras on a commutative body is reduced to the study of simple central algebras on a commutative body.        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4Here we are going to study the simple central algebras (or simple central algebras) on K . Examples  K is a simple central algebra on K . For any whole n \u2265 1, algebra M n ( K ) square matrices is a simple central algebra. A K – Algebra (associative) with division [ first ] central (of finished dimension) is a simple central algebra. These algebras are the overflows D of K whose center is K and which are of finished dimension on K . Either D central division algebra on K . So, for any whole n \u2265 1, algebra M n ( D ) Square matrices of order n on D is a simple central algebra on K . More intrinsically, for any vector space AND finished dimension on D , End algebra ( AND ) Endomorphisms of AND is a simple central algebra on K . It is said that a simple central algebra on K East deployed ( split in English) if there is an integer n \u2265 1 such that A is isomorphic to algebra M n ( K ) Square order matrices n . Either A an algebra on K . It is equivalent to say that:        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4A is a simple central algebra on K ; A is isomorphic to M n ( D ), Or n \u2265 1 and D is an algebra with central division on K ; A is isomorphic to end algebra ( AND ) Endomorphisms of AND , Or AND is a non -zero finite dimension vector space on D , Or D is an algebra with central division on K . Be D And D ‘Central division algebras on K , n And n ‘integers \u2265 1. so that algebra M n ( D ) And M n ‘ ( D ‘) are isomorphic, it is necessary and it is enough that n = n ‘and that algebra D And D ‘are isomorphic.Either AND And AND ‘Vector spaces of non -zero finished dimensions on D And D ‘ respectively. So that ends ends D ( AND ) and ends D ‘ ( AND ‘) on K are isomorphic, it is necessary and it is enough that the algebra bodies D And D ‘ on K are isomorphic and that the dimensions of AND And AND ‘are equal. The classification of simple central algebras on K is therefore reduced to the classification of central division on division on K . Either A Simple central algebra and M A finished type module on A . Then K -Alg\u00e8bre of the endomorphisms of M is a simple central algebra on K . Examples of simple central algebras on certain commutative bodies [ modifier | Modifier and code ] On the body R Real numbers, simple central algebras are, with an isomorphism, the following: the algebras of the form M n ( R ) and form M n ( H ), Or H is the body of quaternions. More intrinsically, these are the algebras of the endomorphisms of real vector spaces and quaternionians of non -zero finished dimensions. And K is an algebraically closed body (for example if K is the body C with complex names) ou yes K is a finished body, the simple central algebras are, with a nearest isomorphisms, those of the form M n ( K ). Properties of simple power plants [ modifier | Modifier and code ] Simple central algebras on K enjoy several remarkable properties. The opposite algebra of a simple central algebra on K is a simple central algebra. The tensorial product of two (or more generally of a finished family) of simple power plants on central K is a simple central algebra. And L is a commutative overflow of K , then algebra L \u2297 K A on L deducted from A by extension of scalars of K To L is a simple central algebra. Be A a simple central algebra on K And B simple algebra on K . Whatever homorphisms (unit) f And g of A In B , there is an invertible element b of B such as g ( x ) = bf ( x ) b \u22121 , for any element x From A ( f And g thereby are conjugated ). In particular, any autumorphism of A is an inner self A , that is to say that it is in the form x \u21a6 axa \u22121 , Or a is an invertible element of A , and this autumorphism is then noted int a . L’application a \u21a6 Int a of the group A * Reversible elements of A In the AUT group K ( A ) autumorphisms of K -algebra A is surjective, and its nucleus is the group K * non -zero scalars of K , and we thus obtain an isomorphism of groups of A * \/ K * on AUT K ( A ). Either D central division algebra on K And AND A finite dimension vector space n on K . Then the group of reversible END elements ( AND ) is the linear group GL ( AND ) of AND , and the application f \u21a6 Int f of GL ( AND ) in Aut K (End( AND )) is a core surjective homomorphism K , and we thus obtain an isomorphism of GL ( AND )\/ K * on AUT K (End( AND )). And n \u2265 2, then the group GL ( AND )\/ K * is canonically isomorphic to the projective group of the projective space  P ( AND ). Either A a simple central algebra on K . Then the dimension of A on K is a square d 2 , and we call degree of A the natural whole d . Be A a simple central algebra on K And d the degree of A . There is a commutative overflow L of K such as the L -Pent central alternate L \u2297 K A on L deducted from A by extension of scalars of K To L is deployed, that is to say isomorphic to M d ( L ), and it is said that such an overflow L of K is a neutralizing body or one deployment of A . Examples  And A is deployed, then K is a neutralizing body of A . Any overly enclosed overorps of L (an algebraic fence of K for example) is a neutralizing body of A . For example, if K is the body R real numbers, C is a neutralizing body of A . There is a neutralizing body L of A tel that the dimension of L is finished, and such that L (considered an extension of K ) is Galoisienne. Either D central division algebra on K . So there is a maximum element L for the inclusion relationship of all the sub-bodys of D which are commutative. SO L is a neutralizing body of D , and more generally M n ( D ). So, for any vector space AND finished dimension on D , L is a neutralizing body of end ( AND ). With an element of a simple central algebra, we can associate scalars which generalize the trace, the determinant, and a polynomial which generalizes the characteristic polynomial, square matrices and endomorphisms of vector space on a commutative body. Be A a simple central algebra on K , d the degree of A , L a neutralizing body of A And B = L \u2297 K A the L -Al\u00e8bre simple central deducted from A by extension of scalars of K To L . For any element x of A and for any isomorphism of L -Alg\u00e8bres h of B on M d ( L ), the trace, the determinant and the polynomial characteristic of the matrix h (1 \u2297 x ) of M d ( L ) only depend on A and of x (and not L or of h ), and they are called reduced trace , reduced standard And reduced characteristic polynomial of x In A (on K ), and we note them TRD A \/ K ( x ), GDR A \/ K ( x ) and PRD A \/ K ( x ) respectively. For example, if A = M d ( K ) or A = End K ( AND ), where E is a non -zero finished vector space on K , the reduced trace, the reduced standard and the characteristic polynomial reduced by an element of A are none other than its trace, its determinant and its characteristic polynomial. In general : Function x \u21a6 Trd A \/ K ( x ) of A In K is a linear form not identically zero in the vector space A . Whatever the elements a And b of A , we have nd A \/ K ( ab ) = GDR A \/ K ( a ) GDR A \/ K ( b ). So that an element a of A be reversible in A , it is necessary and it is enough that NRD A \/ K ( a ) be non -zero. Function x \u21a6 GDR A \/ K ( x ) of the group A * Reversible elements of A In K * is a homomorphism of groups, not necessarily surjective. (He is overjective if A is deployed.) For any element a of A and for any element k of K , we have nd A \/ K ( the ) = k d East Germany A \/ K ( a ). If the body K is infinite, then the function x \u21a6 GDR A \/ K ( x ) of A In K is a homogeneous degree polynomial function d . The degree of the polynomial characteristic of an element of a of A is equal to d , the reduced trace of a is the coefficient of X n – first and the reduced standard of a is the constant term, multiplied by (\u20131) d . Trace and determining of an endomorphism of a quaternionnian vector space [ modifier | Modifier and code ] Either AND A finite dimension vector space n on the body H Quaternions. Then the degree of A = End H ( AND ) is 2 d . By restriction of scalars, we can consider AND Like a complex vector space AND 0 , and then end H ( AND ) is a real unitary sub-algebra of the simple END complex central algebra C ( AND 0 ). For all endomorphism f of AND , the reduced trace, the reduced standard and the characteristic polynomial reduced by the element f of A is none other than the trace, The norme [Ref. necessary] and the characteristic polynomial of the element f The End C ( AND 0 ). Either f Endomorphism of AND . We call trace of f And we note TR f the reduced trace of f , divided by 2. The reduced standard of f  is a real or zero real number [Ref. necessary] , and then we call determining of f an on note that f the square root of the reduced standard of f . N. Bourbaki, Algebra , chapter 8 (in) Thomas W. Hungerford\u00a0 (in) , Algebra , Springer-Verlag (in) Nathan Jacobson, Basic Algebra II , W. H. Freeman, New York, 1989 (in) Max-Albert Kusus (of) , Alexander Mercury, Markus Rost (of) and Jean-Pierre Tignol, The Book of Involutions , AMS, 1998 Semi-summary ring       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/all2en\/wiki32\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/all2en\/wiki32\/algebra-simple-wikipedia\/#breadcrumbitem","name":"algebra simple \u2014 wikipedia"}}]}]