# Rational function body

A rational function field is a term from the mathematical branch of algebra . This object has the algebraic structure of a body .

## definition

The rational function field is the quotient field of the polynomial ring over a field . The construction of is analogous to that of the rational numbers from the integers. The elements can thus be written as with polynomials , which is not the zero polynomial. ${\ displaystyle K (X)}$ ${\ displaystyle K [X]}$ ${\ displaystyle K}$${\ displaystyle K (X)}$${\ displaystyle r \ in K (X)}$${\ displaystyle r = {\ tfrac {f} {g}}}$${\ displaystyle f, g \ in K [X]}$${\ displaystyle g}$

## Notes and properties

The naming is traditional, but should be used with a little caution:

• First, one has to consider the differences between polynomials and polynomial functions. Every polynomial induces a polynomial function, but the polynomial assignment to the polynomial function is only injective if the body is infinite. Example: If the body has 2 elements , then induce and perform the same function . Nevertheless, as elements of the rational body of functions, they are not the same.${\ displaystyle \ rightarrow}$${\ displaystyle K}$${\ displaystyle K = \ mathbb {F} _ {2}}$${\ displaystyle X}$${\ displaystyle X ^ {2}}$${\ displaystyle K}$
• Second, the denominator usually has zeros. Accordingly, the rational function is not entirely defined, but only on a subset that is open to Zariski .${\ displaystyle g}$${\ displaystyle K}$

Example: For is considered a rational function in the sense of the definition above - but the domain is empty. ${\ displaystyle K = \ mathbb {F} _ {3}}$${\ displaystyle {\ tfrac {1} {X ^ {3} -X}}}$${\ displaystyle K}$

The expansion of the body is purely transcendent and therefore especially infinite. With the help of the generalized partial fraction decomposition, it is even possible to specify a base of the vector space . ${\ displaystyle K (X) / K}$${\ displaystyle K}$${\ displaystyle K}$${\ displaystyle K (X)}$

## In several variables

### definition

The rational function field in the variables is defined analogously as the quotient field of the polynomial ring . ${\ displaystyle \ displaystyle K (X_ {1}, \ ldots, X_ {n})}$${\ displaystyle \ displaystyle X_ {1}, \ ldots, X_ {n}}$${\ displaystyle \ displaystyle K [X_ {1}, \ ldots, X_ {n}]}$

### construction

The rational function field can be constructed by successively adjointing a variable and then forming the quotient field. So: ${\ displaystyle \ displaystyle X_ {i}}$

${\ displaystyle \ displaystyle K (X_ {1}, \ ldots, X_ {n})}$is the quotient field of the polynomial ring , i.e. the polynomial ring above the body in the variable${\ displaystyle \ displaystyle K (X_ {1}, \ ldots, X_ {n-1}) [X_ {n}]}$${\ displaystyle \ displaystyle K (X_ {1}, \ ldots, X_ {n-1})}$${\ displaystyle \ displaystyle X_ {n}}$

## Function fields in algebraic geometry

In algebraic geometry , function fields of affine varieties are considered: Let the field be algebraically closed and an affine variety im . Then the ideal is a prime ideal in the polynomial ring , which is why the coordinate ring , ie the quotient ring , is a domain of integrity . ${\ displaystyle K}$ ${\ displaystyle V}$${\ displaystyle K ^ {n}}$${\ displaystyle I (V)}$${\ displaystyle K [X_ {1}, \ ldots, X_ {n}]}$ ${\ displaystyle K [V]}$ ${\ displaystyle K [X_ {1}, \ ldots, X_ {n}] / I (V)}$

The quotient field of the coordinate ring is then called the function field of . Its elements are called rational functions on and may (not empty) open subsets of fact as functions to be considered. ${\ displaystyle K (V)}$${\ displaystyle K [V]}$${\ displaystyle V}$${\ displaystyle V}$${\ displaystyle V}$