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 extensionL/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]
ground, field, mathematics, ground, field, field, fixed, beginning, discussion, contents, linear, algebra, algebraic, geometry, theory, galois, theory, diophantine, geometry, notesuse, editit, used, various, areas, algebra, linear, algebra, edit, linear, algeb. In mathematics a ground field is a field K fixed at the beginning of the discussion Contents 1 Use 1 1 In linear algebra 1 2 In algebraic geometry 1 3 In Lie theory 1 4 In Galois theory 1 5 In Diophantine geometry 2 NotesUse EditIt is used in various areas of algebra In linear algebra Edit In linear algebra the concept of a vector space may be developed over any field In algebraic geometry Edit In algebraic geometry in the foundational developments of Andre 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 Edit 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 Edit 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 Edit 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 Edit Abstract algebraic geometry Encyclopedia of Mathematics EMS Press 2001 1994 Form of an algebraic group Encyclopedia of Mathematics EMS Press 2001 1994 Retrieved from https en wikipedia org w index php title Ground field amp oldid 964393825, wikipedia, wiki, book, books, library,