TITLE: Symmetric power series over $\mathbb{F}_2$
QUESTION [5 upvotes]: Consider the symmetric power series 
$$f = \prod_{i \in I}\left(1+x_i+x_i^2+x_i^4+x_i^8 + x_i^{16} +\ldots \right)$$
in variables $(x_i)_{i \in I}$ over $\mathbb F_2$. Fix some degree $r$, smaller than the number of variables, and denote the degree $r$ part of $f$ by $f^{(r)}$.
From looking at small-dimensional examples, I got the impression that the following might be true: If $r$ is odd, then $f^{(r)}$ is divisible by $\sigma_1 = \sum_{i \in I} x_i$. Could somebody provide a proof or counterexample of that statement? It would be interesting to me knowing the answer, either way.

REPLY [5 votes]: Let $I=\{i_0\} \cup J$ and $g(x)=1+x+x^2+x^4+\dots$.
We show that adding the relation $\sum_{i\in I}x_i=0$, i.e. $x_{i_0}=\sum_{i\ne i_0}x_i$ leads to an even degree power series: 
$$ f=g(x_{i_0})\prod_{j\in J} g(x_j) =\left(1+\sum_{i\in J} (g(x_i)-1)\right)\prod_{j\in J} g(x_j)\\ = \prod_{j\in J} g(x_j) + \sum_{i\in J}\left(g(x_i)-1\right)g(x_i)\prod_{j\in J\setminus \{i\}}g(x_j)\\ =  \prod_{j\in J} g(x_j) + \sum_{i\in J}x_i\prod_{j\in J\setminus \{i\}}g(x_j)= \prod_{j\in J} g(x_j) + \sum_{i\in J}x_i\frac{\partial}{\partial x_i}\prod_{j\in J}g(x_j)$$
Now this is an even degree power series: check that each monomial with an odd number of $x_j^1$ cancels.