I want to show that the power series $$g(x) = \log(1+x) = x - \frac 1 2 x^2 + \frac 1 3 x^3 - \frac 1 4 x^4 + \dotsc$$ is not algebraic. By definition, $g$ is algebraic if there exists a non-zero polynomial $P \in \mathbb C[x,y]$ with $$P(x, g(x)) = 0. \tag{1} \label{eqn1}$$ I know that this can be done using a theorem by Eisenstein¹, but I came up with a different proof:
Suppose $g$ is algebraic, with polynomial $P$ as in \eqref{eqn1}. Wlog we may assume that $P$ is irreducible. Then the inverse function $f(x) = e^x - 1$ is also algebraic, because $$\tag{2} \label{eqn}P(f(y),y) = P(f(y), g(f(y))) = 0.$$ In particular, the graph $\Gamma = \{(f(y), y) : y \in \mathbb C\}$ of $f$ is contained in the algebraic curve $$C = \{(x,y) \in \mathbb C^2 : P(x,y) = 0\} \supset \Gamma.$$ However, the graph $\Gamma$ contains all the points $(0, 2\pi i \cdot k)$ for $k \in \mathbb Z$. This can only happen if $C = \{x = 0\}$ and $P = x$, which is a contradiction to \eqref{eqn}.
Is that proof sound, or did I miss something?
¹ See wikipedia or Pólya and Szegö's Problems and Theorems in Analysis II: If a rational power series $$a_0 + a_1 z + a_2 z^2 + \dotsc$$ represents an algebraic function, then there is a number $T \in \mathbb N$ such that $$a_1 + a_1 Tz + a_2 T^2 z^2 + \dotsc$$ has integer coefficients. In particular, the denominators of the $a_i \in \mathbb Q$ can only have finitely many prime factors.