My question really is:


if $e^{2\pi i\* g(\theta)}$ is an algebraic function in the variable $e^{2 \pi i \theta}$, what restrictions can we put on g? 


My first guess is to say that g is the map that sends everything to zero, or $g(\theta)=n\theta +c$, in which case $e^{2\pi i\* g(\theta)}=1$ or $C\*(e^{2 \pi i \theta})^{n}$ respectivley. 


It seems believable that these would be the only two polynomials, or even rational (or even algebraic?!?) functions that could play the role of g...could a transcendental function do the job of g too?


Any thoughts on where to look? 


P.S. I am afraid that I may not be posing the question well...apologies. 



Looking for Direction on algebraic and transcendental functions
(You should change the tag, this has nothing to do with functional analysis)


By Ax's theorem giving a function field analogue of Schanuel's conjecture, if $g(\theta)$ is an algebraic function of $\theta$ such that $e^{2\pi i \theta}$ and $e^{2 \pi i g(\theta)}$ are algebraically dependent, then $\theta$ and $g(\theta)$ are linearly dependent over $\mathbb{Q}$, so you are almost correct.


<http://en.wikipedia.org/wiki/Schanuel%27s_conjecture>


