Full paper in PDF:
$%T. Recio and J. R.
Sendra, A really elementary proof of real Lüroth’s Theorem, Rev. Mat. Univ. Complut. Madrid
10 (1997), supplementary, 283–290.
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Dpto. de Matemáticas Universidad de Cantabria Santander 39071 — Spain | Dpto. de Matemáticas Universidad de Alcalá Madrid 28871 — Spain |
Classical Lüroth theorem states that every subfield of , where is a transcendental element over , such that strictly contains , must be , for some non constant element in . Therefore, is -isomorphic to . This result can be proved with elementary algebraic techniques, and therefore it is usually included in basic courses on field theory or algebraic curves. In this paper we study the validity of this result under weaker assumptions: namely, if is a subfield of and strictly contains ( the real field, the complex field), when does it hold that is isomorphic to ? Obviously, a necessary condition is that admits an ordering. Here we prove that this condition is also sufficient, and we call such statement the Real Lüroth’s Theorem. There are several ways of proving this result (Riemann’s theorem, Hilbert-Hurwitz (1890)), but we claim that our proof is really elementary, since it does require just same basic background as in the classical version of Lüroth’s.
1991 Mathematics Subject Classification: 14H05, 14P05.