# 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.

## 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. - Second, the denominator usually has zeros. Accordingly, the rational function is not entirely defined, but only on a subset that is open to Zariski .

Example: For is considered a rational function in the sense of the definition above - but the domain is empty.

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 .

## In several variables

### definition

The rational function field in the variables is defined analogously as the quotient field of the polynomial ring .

### construction

The rational function field can be constructed by successively adjointing a variable and then forming the quotient field. So:

- is the quotient field of the polynomial ring , i.e. the polynomial ring above the body in the variable

## 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 .

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.

## literature

- Siegfried Bosch : Algebra . 8th edition. Springer Spectrum, Berlin 2013, ISBN 978-3-642-39566-6 , p. 63 , doi : 10.1007 / 978-3-642-39567-3 ( and ).
- Klaus Hulek : Elementary Algebraic Geometry . 2nd Edition. Springer Spectrum, Wiesbaden 2012, ISBN 978-3-8348-1964-2 , p. 41 , doi : 10.1007 / 978-3-8348-2348-9 (algebraic geometry).