Classical Lüroth theorem states that every subfieldof, whereisa transcendental element over, such thatstrictly contains, mustbe, for some non constant elementin. Therefore,is-isomorphic to. This result can be proved with elementaryalgebraic techniques, and therefore it is usually included in basic courses on fieldtheory or algebraic curves. In this paper we study the validity of this resultunder weaker assumptions: namely, ifis a subfield ofandstrictlycontains(the real field,the complex field), when does it hold thatis isomorphic to? Obviously, a necessary condition is thatadmitsan ordering. Here we prove that this condition is also sufficient, and we callsuch statement the Real Lüroth’s Theorem. There are several ways of provingthis result (Riemann’s theorem, Hilbert-Hurwitz (1890)), but we claim that ourproof is really elementary, since it does require just same basic background asin the classical version of Lüroth’s.
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.