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Let p:C→C be a complex polynomial with p(z)≠0 for all z∈C.

Then p extends to a continuous transformation of the Riemann sphere Cˆ=C∪{∞} (and this extension also has no zeros).

Since the Riemann sphere is compact, there is some ε>0 such that |p(z)|≥ε for all z∈C.

Now consider the holomorphic function g:C→C defined by g(z):=1/p(z).

We have |g(z)|≤1/ε for all z∈C.

By Liouville's Theorem, g is constant. Hence p is also constant, as claimed.

I follow everything until the very end. This is a proof by contradiction, how does it contradict things that p is constant? This is going to sound like a really stupid question but couldn't a complex polynomial be a constant function so that any variable you plug in for x gives the same result because of the choice of coefficients? I am having a lot of difficulties in forming thoughts about functions of complex variables, maybe because I am just not used to them yet and so have a hard time being able to visualize them immediately.

Sorry if this sound incredibly dumb. It's just my brain is fried and for some reason I am just not getting this. Also, I am having a bit of a difficulties differentiating (no pun intended) between analytic functions and holomorphic functions.