We know that for algebraic $e^{it\pi}$, $t$ can not be algebraic irrational by Baker's theorem, but his proof is analytic; is there some algebraic understanding for such fact? If $t$ is transcendent, can $t$ be a period? We say $t$ is period, iff $t$ can be expressed as the volume of domain in $R^{n} defined by polynomial inequalities with algebraic coefficients.

The heart of all transcendence proofs is the analytic fact that there is no integer between $0$ and $1$. This is one of the things I learned from Baker's book "Transcendental Number Theory". 

