Function field

( Algebraic ) function body are in mathematics algebraic correspondences of geometric objects. Function fields over finite fields play an important role in algebraic number theory.

Algebraic definition

It is a body. Then is called a transcendental extension field of finite transcendence degree of an ( algebraic ) function body.

The algebraic closure of in is called constant field.

Function body in the strict sense are function fields of transcendence degree 1 over a finite field. Together with algebraic number fields, they form the class of the global body.

Geometric definition

Is a complete algebraic scheme over a field is the name of the stalk of the structure sheaf in the generic point function of the body. It is a function field over in the algebraic sense.

Example

The rational images on a variety form a functional body. It is a rational function field.

  • Body ( algebra)
  • Body theory
  • Algebraic Geometry
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