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Fixed Field
In algebra, the fixed-point subring R^f of an automorphism ''f'' of a ring ''R'' is the subring of the fixed points of ''f'', that is, :R^f = \. More generally, if ''G'' is a group acting on ''R'', then the subring of ''R'' :R^G = \ is called the fixed subring or, more traditionally, the ring of invariants under . If ''S'' is a set of automorphisms of ''R'', the elements of ''R'' that are fixed by the elements of ''S'' form the ring of invariants under the group generated by ''S''. In particular, the fixed-point subring of an automorphism ''f'' is the ring of invariants of the cyclic group generated by ''f''. In Galois theory, when ''R'' is a field and ''G'' is a group of field automorphisms, the fixed ring is a subfield called the fixed field of the automorphism group; see Fundamental theorem of Galois theory. Along with a module of covariants, the ring of invariants is a central object of study in invariant theory. Geometrically, the rings of invariants are the coordinate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Algebra
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic operations other than the standard arithmetic operations, such as addition and multiplication. Elementary algebra is the main form of algebra taught in schools. It examines mathematical statements using variables for unspecified values and seeks to determine for which values the statements are true. To do so, it uses different methods of transforming equations to isolate variables. Linear algebra is a closely related field that investigates linear equations and combinations of them called '' systems of linear equations''. It provides methods to find the values that solve all equations in the system at the same time, and to study the set of these solutions. Abstract algebra studies algebraic structures, which consist of a set of mathemati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Symmetric Group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group \mathrm_n defined over a finite set of n symbols consists of the permutations that can be performed on the n symbols. Since there are n! (n factorial) such permutation operations, the order (number of elements) of the symmetric group \mathrm_n is n!. Although symmetric groups can be defined on infinite sets, this article focuses on the finite symmetric groups: their applications, their elements, their conjugacy classes, a finite presentation, their subgroups, their automorphism groups, and their representation theory. For the remainder of this article, "symmetric group" will mean a symmetric group on a finite set. The symmetric group is important to diverse areas of mathematics such as Galois theory, invariant theory, the re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Ring Of Polynomial Functions
In mathematics, the ring of polynomial functions on a vector space ''V'' over a field ''k'' gives a coordinate-free analog of a polynomial ring. It is denoted by ''k'' 'V'' If ''V'' is finite dimensional and is viewed as an algebraic variety, then ''k'' 'V''is precisely the coordinate ring of ''V''. The explicit definition of the ring can be given as follows. Given a polynomial ring k _1, \dots, t_n/math>, we can view t_i as a coordinate function on k^n; i.e., t_i(x) = x_i where x = (x_1, \dots, x_n). This suggests the following: given a vector space ''V'', let ''k'' 'V''be the commutative ''k''-algebra generated by the dual space V^*, which is a subring of the ring of all functions V \to k. If we fix a basis for ''V'' and write t_i for its dual basis, then ''k'' 'V''consists of polynomials in t_i. If ''k'' is infinite, then ''k'' 'V''is the symmetric algebra of the dual space V^*. In applications, one also defines ''k'' 'V''when ''V'' is defined over some subfield of ''k' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Free Module
In mathematics, a free module is a module that has a ''basis'', that is, a generating set that is linearly independent. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a field in the commutative case), then there exist non-free modules. Given any set and ring , there is a free -module with basis , which is called the ''free module on'' or ''module of formal'' -''linear combinations'' of the elements of . A free abelian group is precisely a free module over the ring \Z of integers. Definition For a ring R and an R- module M, the set E\subseteq M is a basis for M if: * E is a generating set for M; that is to say, every element of M is a finite sum of elements of E multiplied by coefficients in R; and * E is linearly independent: for every set \\subset E of distinct elements, r_1 e_1 + r_2 e_2 + \cdots + r_n e_n = 0_M implies that r_1 = r_2 = \cdots = r_n = 0_R (where 0_M is the zero element of M and 0_R is the zer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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G-module
In mathematics, given a group ''G'', a ''G''-module is an abelian group ''M'' on which ''G'' acts compatibly with the abelian group structure on ''M''. This widely applicable notion generalizes that of a representation of ''G''. Group (co)homology provides an important set of tools for studying general ''G''-modules. The term ''G''-module is also used for the more general notion of an ''R''-module on which ''G'' acts linearly (i.e. as a group of ''R''-module automorphisms). Definition and basics Let G be a group. A left G-module consists of an abelian group M together with a left group action \rho:G\times M\to M such that :g\cdot(a_1+a_2)=g\cdot a_1+g\cdot a_2 for all a_1 and a_2 in M and all g in G, where g\cdot a denotes \rho(g,a). A right G-module is defined similarly. Given a left G-module M, it can be turned into a right G-module by defining a\cdot g=g^\cdot a. A function f:M\rightarrow N is called a morphism of G-modules (or a G-linear map, or a G-homomorphism) if ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Unipotent Group
In mathematics, a unipotent element ''r'' of a ring ''R'' is one such that ''r'' − 1 is a nilpotent element; in other words, (''r'' − 1)''n'' is zero for some ''n''. In particular, a square matrix ''M'' is a unipotent matrix if and only if its characteristic polynomial ''P''(''t'') is a power of ''t'' − 1. Thus all the eigenvalues of a unipotent matrix are 1. The term quasi-unipotent means that some power is unipotent, for example for a diagonalizable matrix with eigenvalues that are all roots of unity. In the theory of algebraic groups, a group element is unipotent if it acts unipotently in a certain natural group representation. A unipotent affine algebraic group is then a group with all elements unipotent. Definition Definition with matrices Consider the group \mathbb_n of upper-triangular matrices with 1's along the diagonal, so they are the group of matrices :\mathbb_n = \left\. Then, a unipotent group can be defined a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Integral Element
In commutative algebra, an element ''b'' of a commutative ring ''B'' is said to be integral over a subring ''A'' of ''B'' if ''b'' is a root of a polynomial, root of some monic polynomial over ''A''. If ''A'', ''B'' are field (mathematics), fields, then the notions of "integral over" and of an "integral extension" are precisely "algebraic element, algebraic over" and "algebraic extensions" in field theory (mathematics), field theory (since the root of any polynomial is the root of a monic polynomial). The case of greatest interest in number theory is that of complex numbers integral over Z (e.g., \sqrt or 1+i); in this context, the integral elements are usually called algebraic integers. The algebraic integers in a finite field extension, extension field ''k'' of the rational number, rationals Q form a subring of ''k'', called the ring of integers of ''k'', a central object of study in algebraic number theory. In this article, the term ''ring (mathematics), ring'' will be underst ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Finitely Generated Algebra
In mathematics, a finitely generated algebra (also called an algebra of finite type) is a commutative associative algebra ''A'' over a field ''K'' where there exists a finite set of elements a_1,\dots,a_n of ''A'' such that every element of ''A'' can be expressed as a polynomial in a_1,\dots,a_n, with coefficients in ''K''. Equivalently, there exist elements a_1,\dots,a_n\in A such that the evaluation homomorphism at =(a_1,\dots,a_n) :\phi_\colon K _1,\dots,X_ntwoheadrightarrow A is surjective; thus, by applying the first isomorphism theorem, A \simeq K _1,\dots,X_n(\phi_). Conversely, A:= K _1,\dots,X_nI for any ideal I\subseteq K _1,\dots,X_n/math> is a K-algebra of finite type, indeed any element of A is a polynomial in the cosets a_i:=X_i+I, i=1,\dots,n with coefficients in K. Therefore, we obtain the following characterisation of finitely generated K-algebras :A is a finitely generated K-algebra if and only if it is isomorphic as a K-algebra to a quotient ring of the type ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Finite Group
In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving transformations. Important examples of finite groups include cyclic groups and permutation groups. The study of finite groups has been an integral part of group theory since it arose in the 19th century. One major area of study has been classification: the classification of finite simple groups (those with no nontrivial normal subgroup) was completed in 2004. History During the twentieth century, mathematicians investigated some aspects of the theory of finite groups in great depth, especially the local theory of finite groups and the theory of solvable and nilpotent groups. As a consequence, the complete classification of finite simple groups was achieved, meaning that all those simple groups from which all finite groups can be bu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Hilbert's Fourteenth Problem
In mathematics, Hilbert's fourteenth problem, that is, number 14 of Hilbert's problems proposed in 1900, asks whether certain algebras are finitely generated. The setting is as follows: Assume that ''k'' is a field and let ''K'' be a subfield of the field of rational functions in ''n'' variables, :''k''(''x''1, ..., ''x''''n'' ) over ''k''. Consider now the ''k''-algebra ''R'' defined as the intersection : R:= K \cap k _1, \dots, x_n\ . Hilbert conjectured that all such algebras are finitely generated over ''k''. Some results were obtained confirming Hilbert's conjecture in special cases and for certain classes of rings (in particular the conjecture was proved unconditionally for ''n'' = 1 and ''n'' = 2 by Zariski in 1954). Then in 1959 Masayoshi Nagata found a counterexample to Hilbert's conjecture. The counterexample of Nagata is a suitably constructed ring of invariants for the action of a linear algebraic group. History The problem originally arose in algebraic inva ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |