Parametrization of Algebraic Curves over Optimal Field Extensions

In this paper we investigate the problem of determining rational parametrizations of plane algebraic curves over an algebraic extension of least degree over the field of definition. This problem reduces to the problem of finding simple points with coordinates in the field of definition on algebraic curves of genus 0. Consequently we are also able to decide parametrizability over the reals. We generalize a classical theorem of Hilbert and Hurwitz about birational transformations. An efficient algorithm for computing such optimal parametrizations is presented.