Let $R: \widehat{\mathbb{C}} \to \widehat{\mathbb{C}}$ be a nonconstant holomorphic self-map of the Riemann sphere. Suppose that, on $\mathbb{C}$, $R$ is written in lowest terms as
and with a [Möbius transformation](/page/M%C3%B6bius%20Transformation) meaning a map $T: \widehat{\mathbb{C}} \to \widehat{\mathbb{C}}$ represented on finite points by
for coefficients $a,b,c,d \in \mathbb{C}$ satisfying $ad-bc \ne 0$, with the usual extended values at the pole and at $\infty$, the map $R$ is a Möbius transformation if and only if $\deg R=1$.