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non-constant element of rational function field
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(Theorem)
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Let $K$ be a field. Every simple transcendent field extension $K(\alpha)/K$ may be represented by the extension $K(X)/K$ , where $K(X)$ is the field of fractions of the polynomial ring $K[X]$ in one indeterminate $X$ . The elements of $K(X)$ are rational functions, i.e. rational expressions
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(1) |
with $f(X)$ and $g(X)$ polynomials in $K[X]$ .
Proof. The element $X$ satisfies the equation
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(2) |
the coefficients of which are in the field $K(\varrho)$ , actually in the ring $K[\varrho]$ . If all these coefficients were zero, we could take one non-zero coefficient $b_\nu$ in $g(X)$ and the coefficient $a_\nu$ of the same power of $X$ in $f(X)$ , and then we would have especially $\varrho b_\nu\!-a_\nu = 0$ ; this would mean that $\varrho =
\frac{a_\nu}{b_\nu}$ = 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 $K(\varrho)$ . If $K(\varrho)$ 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 $K(\varrho)$ is transcendental, Q.E.D.
Further, $X$ is a zero of the $n^\mathrm{th}$ degree polynomial $$h(Y) = \varrho\,g(Y)\!-\!f(Y)$$ of the ring $K(\varrho)[Y]$ , actually of the ring $K[\varrho][Y]$ , i.e. of $K[\varrho$ ,Y]. The polynomial is irreducible in this ring, since otherwise it would have there two factors, and because $h(Y)$ is linear in $\varrho$ , 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 $K(\varrho)$ of degree $n$ and therefore also $$(K(X):K(\varrho)) = n,$$ Q.E.D.
- 1
- B. L. van der Waerden: Algebra. Siebte Auflage der Modernen Algebra. Erster Teil.
-- Springer-Verlag. Berlin, Heidelberg (1966).
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"non-constant element of rational function field" is owned by pahio.
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field of rational functions, rational function field |
This object's parent.
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Cross-references: conclusion, relatively prime, factors, irreducible, mean, power, ring, coefficients, equation, proof, algebraic, base field, transcendental, denominator, numerator, degrees, lowest terms, reduced, polynomials, rational functions, indeterminate, polynomial ring, field of fractions, extension, field extension, field
There are 5 references to this entry.
This is version 14 of non-constant element of rational function field, born on 2005-02-16, modified 2005-08-26.
Object id is 6762, canonical name is NonConstantElementOfRationalFunctionField.
Accessed 4069 times total.
Classification:
| AMS MSC: | 12F99 (Field theory and polynomials :: Field extensions :: Miscellaneous) |
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Pending Errata and Addenda
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