Ohm, Jack On subfields of rational function fields. (English) Zbl 0546.12013 Arch. Math. 42, 136-138 (1984). 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 Cited in 1 ReviewCited in 8 Documents MSC: 12F20 Transcendental field extensions 11R58 Arithmetic theory of algebraic function fields 14G05 Rational points Keywords:purely transcendental extension; degree of transcendency; unirational fields PDFBibTeX XMLCite \textit{J. Ohm}, Arch. Math. 42, 136--138 (1984; Zbl 0546.12013) Full Text: DOI References: [1] J. Deveney, Ruled function fields. Proc. Amer. Math. Soc.86, 213–215 (1982). · Zbl 0529.12016 · doi:10.1090/S0002-9939-1982-0667276-6 [2] M. Nagata, A theorem on valuation rings and its applications. Nagoya Math. J.29, 85–91 (1967). · Zbl 0146.26302 [3] P. Roquette, Isomorphisms of generic splitting fields of simple algebras. J. Reine Angew. Math.214–215, 207–226 (1964). · Zbl 0219.16023 · doi:10.1515/crll.1964.214-215.207 [4] I. R.Shafarevich, Basic algebraic geometry. Grundlehren 213, Heidelberg 1974. · Zbl 0284.14001 [5] O.Zariski and P.Samuel, Commutative algebra II. Princeton 1960. · Zbl 0121.27801 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.