Lüroth field extensions

Abstract: Let K be a field of characteristic zero, and let be a purely transcendental field extension of K of transcendence degree . Lüroth''s Theorem. Let E be a subfield of that properly contains the field K. Then for a transcendental element . A field extension is called a Lüroth field extension if and every subfield with 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 it''s algebraic closure in F then so is a purely transcendental field extension . As a consequence of this result we have a description of integrally closed subalgebras of of dimension 1. Theorem 2. Suppose, in addition, that K is an algebraically closed field. Let R be a K-subalgebra of the field that is integrally closed in and the transcendence degree of its field of fractions is 1 over K. Then there exists a transcendental element over K such that . [Copyright &y& Elsevier]