non-constant element of rational function field
Let be a field. Every simple (http://planetmath.org/SimpleFieldExtension) transcendent field extension may be represented by the extension , where is the field of fractions of the polynomial ring in one indeterminate . The elements of are rational functions, i.e. rational expressions
with and polynomials in .
Proof. The element satisfies the equation
the coefficients of which are in the field , actually in the ring . If all these coefficients were zero, we could take one non-zero coefficient in and the coefficient of the same power of in , and then we would have especially ; this would mean that = constant, contrary to the supposition. Thus at least one coefficient in (2) differs from zero, and we conclude that is algebraic with respect to . If were algebraic with respect to , then also should be algebraic with respect to . This is not true, and therefore we see that is transcendental, Q.E.D.
Further, is a zero of the degree polynomial
of the ring , actually of the ring , i.e. of , Y]. The polynomial is irreducible in this ring, since otherwise it would have there two factors, and because is linear in , the other factor should depend only on ; but there can not be such a factor, for the polynomials and are relatively prime. The conclusion is that is an algebraic element over of degree and therefore also
B. L. van der Waerden: Algebra. Siebte Auflage der Modernen Algebra. Erster Teil.
— Springer-Verlag. Berlin, Heidelberg (1966).
|Title||non-constant element of rational function field|
|Date of creation||2013-03-22 15:02:50|
|Last modified on||2013-03-22 15:02:50|
|Last modified by||pahio (2872)|
|Synonym||field of rational functions|
|Synonym||rational function field|