In mathematics, 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

In linear algebra, the concept of a vector space may be developed over any field.

### In algebraic geometry

In algebraic geometry, in the foundational developments of AndrĂ© 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]}

### In Lie theory

Reference to a ground field may be common in the theory of Lie algebras (*qua* vector spaces) and algebraic groups (*qua* algebraic varieties).

### In Galois theory

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 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.^{[2]}

## Notes

**^**"Abstract algebraic geometry",*Encyclopedia of Mathematics*, EMS Press, 2001 [1994]**^**"Form of an algebraic group",*Encyclopedia of Mathematics*, EMS Press, 2001 [1994]