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 $\displaystyle K = $ the unit circle in the complex plane, and $\displaystyle \mathscr{A} $ the algebra of functions of the form $\displaystyle f(e^{i \theta}) = \displaystyle \sum^{N}_{n = 0} c_{n}e^{in \theta}$ for $\displaystyle \theta \in \mathbb{R}$. Show that $\displaystyle A$ separates points on $\displaystyle K$.

So my thinking: I want to take two points $\displaystyle e^{ia} \ne e^{ib}$ and show that there is a function in $\displaystyle A$ such that $\displaystyle f(e^{ia}) \ne f(e^{ib})$. Well, assuming $\displaystyle N \geq M$, then $\displaystyle \displaystyle \sum^{N}_{n = 0} c_{n}e^{ina} = \sum^{M}_{m = 0} c_{m}e^{ima} \Leftrightarrow c_{1}(e^{ia} - e^{ib}) + ... + c_{N - M + 1}e^{i(N - M)a} + ... + c_{N}e^{iNa} = 0$. So I want to show that this last equality fails.