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.
@article{urn:eudml:doc:44266,
title = {A really elementary proof of real L\"uroth's theorem.},
journal = {Revista Matem\'atica de la Universidad Complutense de Madrid},
volume = {10},
year = {1997},
pages = {283-290},
zbl = {0901.12002},
mrnumber = {MR1485305},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:44266}
}
Recio, T.; Sendra, J. R. A really elementary proof of real Lüroth's theorem.. Revista Matemática de la Universidad Complutense de Madrid, Tome 10 (1997) pp. 283-290. http://gdmltest.u-ga.fr/item/urn:eudml:doc:44266/