# A really elementary proof of real Lüroth's theorem.

• Volume: 10, Issue: Supl., page 283-290
• ISSN: 1139-1138

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## Abstract

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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.

## How to cite

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Recio, T., and Sendra, J. R.. "A really elementary proof of real Lüroth's theorem.." Revista Matemática de la Universidad Complutense de Madrid 10.Supl. (1997): 283-290. <http://eudml.org/doc/44266>.

@article{Recio1997,
abstract = {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.},
author = {Recio, T., Sendra, J. R.},
keywords = {Geometría algebraica; Isomorfismo; Subconjuntos; Teoremas; function fields; transcendental extensions; Lüroth theorem; orderable subfield},
language = {eng},
number = {Supl.},
pages = {283-290},
title = {A really elementary proof of real Lüroth's theorem.},
url = {http://eudml.org/doc/44266},
volume = {10},
year = {1997},
}

TY - JOUR
AU - Recio, T.
AU - Sendra, J. R.
TI - A really elementary proof of real Lüroth's theorem.
PY - 1997
VL - 10
IS - Supl.
SP - 283
EP - 290
AB - 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.
LA - eng
KW - Geometría algebraica; Isomorfismo; Subconjuntos; Teoremas; function fields; transcendental extensions; Lüroth theorem; orderable subfield
UR - http://eudml.org/doc/44266
ER -

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