mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a ground field is a field ''K'' fixed at the beginning of the discussion.

Use

It is used in various areas of algebra:

In linear algebra

linear algebra Linear algebra is the branch of mathematics concerning linear equations such as :a_1x_1+\cdots +a_nx_n=b, linear maps such as :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mathemat ...

, the concept of a

vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...

may be developed over any field.

In algebraic geometry

algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...

, in the foundational developments of

André Weil André Weil (; ; 6 May 1906 – 6 August 1998) was a French mathematician, known for his foundational work in number theory and algebraic geometry. He was one of the most influential mathematicians of the twentieth century. His influence is du ...

the use of fields other than the

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...

s was essential to expand the definitions to include the idea of abstract algebraic variety over ''K'', and

generic point In algebraic geometry, a generic point ''P'' of an algebraic variety ''X'' is a point in a ''general position'', at which all generic property, generic properties are true, a generic property being a property which is true for Almost everywhere, ...

relative to ''K''.

In Lie theory

Reference to a ground field may be common in the theory of

Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...

s (''qua'' vector spaces) and algebraic groups (''qua'' algebraic varieties).

In Galois theory

Galois theory In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field (mathematics), field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems ...

, given a

field extension In mathematics, particularly in algebra, a field extension is a pair of fields K \subseteq L, such that the operations of ''K'' are those of ''L'' restricted to ''K''. In this case, ''L'' is an extension field of ''K'' and ''K'' is a subfield of ...

''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 In mathematics, specifically algebraic geometry, a scheme is a structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations and define the same algebraic variety but different s ...

, 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 Diophantine geometry

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.

Notes

{{reflist Field (mathematics) *