Function body

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

Algebraic definition

It is a body . Then a transcendent extension field of finite degree of transcendence is called an (algebraic) function field. ${\displaystyle k}$ ${\displaystyle K/k}$

The algebraic closure of in is called a constant field . ${\displaystyle k}$ ${\displaystyle K}$

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

Geometric definition

If a whole algebraic scheme is over a body , then the stalk of the structural thread in the generic point is called the function body of . It is a function body over in the algebraic sense. ${\displaystyle X}$ ${\displaystyle k}$ ${\displaystyle X}$ ${\displaystyle k}$

example

The rational functions on an (irreducible) variety form a function body.