non-constant element of rational function field

Let K be a field.   Every simple ( transcendent field extension K(α)/K may be represented by the extension K(X)/K, where K(X) is the field of fractionsMathworldPlanetmath of the polynomial ringMathworldPlanetmath K[X] in one indeterminateMathworldPlanetmath X.  The elements of K(X) are rational functions, i.e. rational expressions

ϱ=f(X)g(X) (1)

with f(X) and g(X) polynomialsMathworldPlanetmathPlanetmath in K[X].


Let the non-constant rational function (1) be reduced to lowest terms and let the greater of the degrees of its numerator and denominator be n.  This element ϱ is transcendentalPlanetmathPlanetmath with respect to the base fieldMathworldPlanetmath K.  The field extension K(X)/K(ϱ) is algebraic and of degree n.

Proof.  The element X satisfies the equation

ϱg(X)-f(X)=0, (2)

the coefficients of which are in the field K(ϱ), actually in the ring K[ϱ].  If all these coefficients were zero, we could take one non-zero coefficient bν in g(X) and the coefficient aν of the same power of X in f(X), and then we would have especially  ϱbν-aν=0;  this would mean that  ϱ=aνbν = constant, contrary to the supposition.  Thus at least one coefficient in (2) differs from zero, and we conclude that X is algebraic with respect to K(ϱ).  If K(ϱ) were algebraic with respect to K, then also X should be algebraic with respect to K.  This is not true, and therefore we see that K(ϱ) is transcendental, Q.E.D.

Further, X is a zero of the nth degree polynomial


of the ring K(ϱ)[Y], actually of the ring K[ϱ][Y], i.e. of K[ϱ, Y].  The polynomial is irreducible in this ring, since otherwise it would have there two factors, and because h(Y) is linear in ϱ, the other factor should depend only on Y; but there can not be such a factor, for the polynomials f(Z) and g(Z) are relatively prime.  The conclusion is that X is an algebraic element over K(ϱ) of degree n and therefore also




  • 1 B. L. van der Waerden: AlgebraMathworldPlanetmathPlanetmathPlanetmath.  Siebte Auflage der Modernen Algebra.  Erster Teil.
    — Springer-Verlag. Berlin, Heidelberg (1966).
Title non-constant element of rational function field
Canonical name NonconstantElementOfRationalFunctionField
Date of creation 2013-03-22 15:02:50
Last modified on 2013-03-22 15:02:50
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 17
Author pahio (2872)
Entry type Theorem
Classification msc 12F99
Synonym field of rational functions
Synonym rational function field