TITLE: Is the set of roots of power series with integer coefficients countable?
QUESTION [13 upvotes]: Title says it all.  It is clear that the set of all power series with integer coefficients is not countable while the set of algebraic numbers is countable, but I do not know whether the set of roots for power series with integer roots is uncountable.

REPLY [14 votes]: Let $0<a<1$ be irrational. I claim that there is a power series
$$\sum{a_nx^n}$$
 with integer coefficients that vanishes at $a$.
If $a>\frac12$, let $a_0=1$ and $a_1=-1$, otherwise let $a_0=0$ and $a_1=1$. Suppose recursively that $k\ge 1$ we have defined $a_0,\ldots,a_N$ such that
$$0<\sum_{n=1}^N{a_na^n}<\frac1{k+1}$$
We choose $N'$ and $a_{N+1},\ldots,a_{N'}$ such that
$$0<\sum_{n=1}^{N'}{a_na^n}<\frac1{k+2}$$
If the sum (let's call it $A$ for brevity) already satisfies this let $N'=N$. Otherwise choose $N'$ large enough that 
$$a^{N'}<\frac1{2(k+2)}$$
Let $a_i=0$ if $N<i<N'$. Then let
$$a_{N'}=-\left\lceil\frac{(A-\frac1{k+2})}{a^{N'}}\right\rceil$$
Then
$$0<A+a_{N'}a^{N'}<\frac1{k+2}$$
Letting $k\to\infty$, we end up with a power series with integer coefficients that vanishes at $a$, which should answer your question.