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. ${\displaystyle K(V)}$ ${\displaystyle K[V]}$ ${\displaystyle V}$ ${\displaystyle V}$ ${\displaystyle V}$

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 ). ${\displaystyle K(X)}$ ${\displaystyle K[X_{1},\ldots ,X_{n}]}$
• 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).