How to show $e^{2 \pi i \theta}$ is not algebraic.

Question

I was wondering if someone could possibly help me figure out how to show $e^{2 \pi i \theta}$ is not algebraic when $\theta$ is irrational.

Thanks!

Answer 1

Note that as stated the result is not correct. For example, there is a real number $t$ such that $\cos(2\pi t)$ is equal to $\dfrac{3}{5}$, and it can be shown that $t$ is not rational. But if $\theta$ is an irrational algebraic number, then indeed $e^{2\pi i\theta}$ is transcendental.

Answer 2

Use Lindemann–Weierstrass theorem

http://en.wikipedia.org/wiki/Lindemann%E2%80%93Weierstrass_theorem