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On subfields of rational function fields. (English) Zbl 0546.12013

Let \(k\subset K\) and \(k\subset L\) be field extensions. Suppose that \(L\) is contained in a purely transcendental extension of \(K\). Then \(L\) is \(k\)-isomorphic to a subfield of \(K\), provided the degree of transcendency of \(L| k\) satisfies: \(dt L| k\leq dt K| k.\) This statement constitutes as useful lemma in certain investigations of unirational fields. Until now, however, it was proved in the literature for infinite base field \(k\) only. The proof presented here is valid also for finite base field.
Reviewer: P.Roquette

MSC:

12F20 Transcendental field extensions
11R58 Arithmetic theory of algebraic function fields
14G05 Rational points
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References:

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