Lüroth's Theorem

If $x$ and $y$ are nonconstant rational functions of a parameter, the curve so defined has Genus 0. Furthermore, $x$ and $y$ may be expressed rationally in terms of a parameter which is rational in them.

References

Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 246, 1959.