In mathematics, a ground field is a field K fixed at the beginning of the discussion.
Mathematics includes the study of such topics as quantity, structure, space, and change.
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure, which is widely used in algebra, number theory and many other areas of mathematics.
It is used in various areas of algebra:
In linear algebra, the concept of a vector space may be developed over any field.
Linear algebra is the branch of mathematics concerning linear equations such as
A vector space is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars. Scalars are often taken to be real numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called axioms, listed below.
In algebraic geometry, in the foundational developments of André Weil the use of fields other than the complex numbers was essential to expand the definitions to include the idea of abstract algebraic variety over K, and generic point relative to K. [1]
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros.
André Weil was an influential French mathematician of the 20th century, known for his foundational work in number theory and algebraic geometry. He was a founding member and the de facto early leader of the mathematical Bourbaki group. The philosopher Simone Weil was his sister.
A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is a solution of the equation x2 = −1. Because no real number satisfies this equation, i is called an imaginary number. For the complex number a + bi, a is called the real part, and b is called the imaginary part. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers, and are fundamental in many aspects of the scientific description of the natural world.
Reference to a ground field may be common in the theory of Lie algebras (qua vector spaces) and algebraic groups (qua algebraic varieties).
In mathematics, a Lie algebra is a vector space together with a non-associative, alternating bilinear map , called the Lie bracket, satisfying the Jacobi identity.
In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inversion operations are given by regular maps on the variety.
In Galois theory, given a field extension L/K, the field K that is being extended may be considered the ground field for an argument or discussion. Within algebraic geometry, from the point of view of scheme theory, the spectrum Spec(K) of the ground field K plays the role of final object in the category of K-schemes, and its structure and symmetry may be richer than the fact that the space of the scheme is a point might suggest.
In mathematics, Galois theory provides a connection between field theory and group theory. Using Galois theory, certain problems in field theory can be reduced to group theory, which is in some sense simpler and better understood.
In mathematics, and, in particular, in algebra, a field extension is a pair of fields such that the operations of E are those of F restricted to E. In this case, F is an extension field of E and E is a subfield of F. For example, under the usual notions of addition and multiplication, the complex numbers are an extension field of the real numbers; the real numbers are a subfield of the complex numbers.
In diophantine geometry the characteristic problems of the subject are those caused by the fact that the ground field K is not taken to be algebraically closed. The field of definition of a variety given abstractly may be smaller than the ground field, and two varieties may become isomorphic when the ground field is enlarged, a major topic in Galois cohomology. [2]
In mathematics, Diophantine geometry is the study of points of algebraic varieties with coordinates in the integers, rational numbers, and their generalizations. These generalizations typically are fields that are not algebraically closed, such as number fields, finite fields, function fields, and p-adic fields. It is a sub-branch of arithmetic geometry and is one approach to the theory of Diophantine equations, formulating questions about such equations in terms of algebraic geometry.
In mathematics, the field of definition of an algebraic variety V is essentially the smallest field to which the coefficients of the polynomials defining V can belong. Given polynomials, with coefficients in a field K, it may not be obvious whether there is a smaller field k, and other polynomials defined over k, which still define V.
In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups. A Galois group G associated to a field extension L/K acts in a natural way on some abelian groups, for example those constructed directly from L, but also through other Galois representations that may be derived by more abstract means. Galois cohomology accounts for the way in which taking Galois-invariant elements fails to be an exact functor.
In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois group is also cyclic, the extension is also called a cyclic extension. Going in the other direction, a Galois extension is called solvable if its Galois group is solvable, i.e., if the group can be decomposed into a series of normal extensions of an abelian group.
In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions. Abelian varieties are at the same time among the most studied objects in algebraic geometry and indispensable tools for much research on other topics in algebraic geometry and number theory.
In abstract algebra, the theory of fields lacks a direct product: the direct product of two fields, considered as a ring is never itself a field. Nonetheless, it is often required to "join" two fields K and L, either in cases where K and L are given as subfields of a larger field M, or when K and L are both field extensions of a smaller field N.
In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of algebraic varieties.
In mathematics, the canonical bundle of a non-singular algebraic variety of dimension over a field is the line bundle , which is the nth exterior power of the cotangent bundle Ω on V.
In mathematical logic, Morley rank, introduced by Michael D. Morley (1965), is a means of measuring the size of a subset of a model of a theory, generalizing the notion of dimension in algebraic geometry.
In mathematics, specifically complex geometry, a complex-analytic variety is defined locally as the set of common zeros of finitely many analytic functions. It is analogous to the included concept of complex algebraic variety, and every complex manifold is an analytic variety. Since analytic varieties may have singular points, not all analytic varieties are complex manifolds. A further generalization of analytic varieties is provided by the notion of complex analytic space. An analytic variety is also called a analytic set.
In mathematics, the unitarian trick is a device in the representation theory of Lie groups, introduced by Adolf Hurwitz (1897) for the special linear group and by Hermann Weyl for general semisimple groups. It applies to show that the representation theory of some group G is in a qualitative way controlled by that of some other compact group K. An important example is that in which G is the complex general linear group, and K the unitary group acting on vectors of the same size. From the fact that the representations of K are completely reducible, the same is concluded for those of G, at least in finite dimensions.
In mathematics, a Severi–Brauer variety over a field K is an algebraic variety V which becomes isomorphic to a projective space over an algebraic closure of K. The varieties are associated to central simple algebras in such a way that the algebra splits over K if and only if the variety has a point rational over K. Francesco Severi (1932) studied these varieties, and they are also named after Richard Brauer because of their close relation to the Brauer 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 mathematics, formal moduli are an aspect of the theory of moduli spaces, closely linked to deformation theory and formal geometry. Roughly speaking, deformation theory can provide the Taylor polynomial level of information about deformations, while formal moduli theory can assemble consistent Taylor polynomials to make a formal power series theory. The step to moduli spaces, properly speaking, is an algebraization question, and has been largely put on a firm basis by Artin's approximation theorem.
In mathematics, a moduli scheme is a moduli space that exists in the category of schemes developed by Alexander Grothendieck. Some important moduli problems of algebraic geometry can be satisfactorily solved by means of scheme theory alone, while others require some extension of the 'geometric object' concept.
In mathematics, a free Lie algebra, over a given field K, is a Lie algebra generated by a set X, without any imposed relations other than the defining relations of alternating bilinearity and the Jacobi identity.
In abstract algebra and functional analysis, Baer rings, Baer *-rings, Rickart rings, Rickart *-rings, and AW*-algebras are various attempts to give an algebraic analogue of von Neumann algebras, using axioms about annihilators of various sets.
In mathematics, a Riemann form in the theory of abelian varieties and modular forms, is the following data:
In mathematics, an algebraic differential equation is a differential equation that can be expressed by means of differential algebra. There are several such notions, according to the concept of differential algebra used.
In abstract algebra, a commutant-associative algebra is a nonassociative algebra over a field whose multiplication satisfies the following axiom:
In mathematics, the study of special values of L-functions is a subfield of number theory devoted to generalising formulae such as the Leibniz formula for pi, namely
In algebra, the Jacobson–Bourbaki theorem is a theorem used to extend Galois theory to field extensions that need not be separable. It was introduced by Nathan Jacobson (1944) for commutative fields and extended to non-commutative fields by Jacobson (1947), and Henri Cartan (1947) who credited the result to unpublished work by Nicolas Bourbaki. The extension of Galois theory to normal extensions is called the Jacobson–Bourbaki correspondence, which replaces the correspondence between some subfields of a field and some subgroups of a Galois group by a correspondence between some sub division rings of a division ring and some subalgebras of an algebra.