Proof that a certain entire function is a polynomial

Solution 1:

Here's a solution avoiding big Picard, but using Casorati-Weierstrass and Baire instead.

Assume that $f$ is not a polynomial. Then $f$ has an essential singularity at $\infty$. Let $k \in \mathbb{N}$ and let $D_k = \{ |z| > k \}$. By Casorati-Weierstrass, $f(D_k)$ is dense in $\mathbb{C}$, and by the open mapping theorem, $f(D_k)$ is open.

Baire's category theorem shows that $\bigcap_k f(D_k)$ is non-empty, so there is some $w \in \mathbb{C}$ (in fact, an open dense set of $w$:s) such that the equation $f(z) = w$ has infinitely many solutions, which is our desired contradiction.