Lüroth field extensions

Let K be a field of characteristic zero, and let K(x1,…,xn) be a purely transcendental field extension of K of transcendence degree ngreater-or-equal, slanted1. Lürothʹs Theorem. Let E be a subfield of K(x1) that properly contains the field K. Then E=K(e) for a transcendental element eset membership, variantE. A field extension F/K is called a Lüroth field extension if tr.degK(F)greater-or-equal, slanted1 and every subfield Ksubset ofEsubset of or equal toF with tr.degK(E)=1 is a rational function field in one variable. In this paper, we prove Theorem 1. If F is a Lüroth field extension of field K of characteristic zero that coincides with itʹs algebraic closure in F then so is a purely transcendental field extension image. As a consequence of this result we have a description of integrally closed subalgebras of K(x1,…,xn) of dimension 1. Theorem 2. Suppose, in addition, that K is an algebraically closed field. Let R be a K-subalgebra of the field K(x1,…,xn) that is integrally closed in K(x1,…,xn) and the transcendence degree of its field of fractions Q(R) is 1 over K. Then there exists a transcendental element xset membership, variantR over K such that K[x]subset of or equal toRsubset of or equal toK(x)=Q(R).