it is a classical fact that free (discrete) groups can be embedded in $gl_{2}(\mathbb{z})$. in 1987, zubkov showed that for a free pro-$p$ group $f_{\hat{p}}$, the situation changes, and when $p>2$, $f_{\hat{p}}$ cannot be embedded in $gl_{2}(\delta)$ when $\delta$ is a profinite ring. in 2005, inspired by kemer's solution to the specht problem, zelmanov sketched a proof for the following generalization: for every $d\in\mathbb{n}$ and large enough prime $p\gg d$, $f_{\hat{p}}$ cannot be embedded in $gl_{d}(\delta)$. the natural question then is: what can be said when $p$ is not large enough? what can be said in the case $d=p=2$ ? in the talk i am going to describe the proof of the following theorem: $f_{\hat{2}}$ cannot be embedded in $gl_{2}(\delta)$ when $char(\delta)=2$. the main idea of the proof is the use of trace identities in order to apply finiteness properties of a noetherian trace ring through the artin-rees lemma (joint with e. zelmanov).