A really elementary proof of real Lüroth's theorem.

Título inglés	A really elementary proof of real Lüroth's theorem.
Título español	Prueba realmente elemental del teorema real de Lüroth.
Autor/es	Recio, T. ; Sendra, J. R.
Organización	Dep. Mat. Univ. Cantabria, Santander, España;Dep. Mat. Univ. Alcalá, Alcalá de Henares (Madrid), España
Revista	0214-3577
Publicación	1997, 10 (Supl.): 283-290, 7 Ref.
Tipo de documento	articulo
Idioma	Inglés
Resumen inglés	Classical Lüroth theorem states that every subfield K of K(t), where t is a transcendental element over K, such that K strictly contains K, must be K = K(h(t)), for some non constant element h(t) in K(t). Therefore, K is K-isomorphic to K(t). 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 K is a subfield of C(t) and K strictly contains R (R the real field, C the complex field), when does it hold that K is isomorphic to R(t)? Obviously, a necessary condition is that K 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.
Clasificación UNESCO	120101
Palabras clave español	Geometría algebraica ; Isomorfismo ; Subconjuntos ; Teoremas
Código MathReviews	MR1485305
Código Z-Math	Zbl 0901.12002
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