A theorem that can be stated either in the language of abstract algebraic curves or transcendental extensions.
For an abstract algebraic curve, if and are nonconstant rational functions of a parameter, the curve so defined has curve genus 0. Furthermore, and may be expressed rationally in terms of a parameter which is rational in them (Coolidge 1959, p. 246).
For simple transcendental extensions, all proper extensions of a field which are contained in a simple transcendental extension of are also simple transcendental. In particular, if is an intermediate field between and the field of rational functions over , then for some nonconstant rational function (van der Waerden 1966, p. 198).