I'm wondering if I'm getting stumped here because of my limited familiarity with complex numbers. So if anybody sees what's going on, if you could answer with a mild hint/nudge rather than a full answer, I would appreciate it.

The problem:

Let the unit circle in the complex plane, and the algebra of functions of the form for . Show that separates points on .

So my thinking: I want to take two points and show that there is a function in such that . Well, assuming , then . So I want to show that this last equality fails.