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

Revista Matemática de la Universidad Complutense de Madrid (1997)

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

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topRecio, 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.},

journal = {Revista Matemática de la Universidad Complutense de Madrid},

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.

JO - Revista Matemática de la Universidad Complutense de Madrid

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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