Let us define the automorphism $\sigma$ on ${\Bbb F}_2[[X,S]]$ such that
$\sigma \colon S \mapsto S + S^2 + S^3$
$\sigma \colon X \mapsto X + S$.
It is easy to see that the ideal $(S)$ is stable by $\sigma$. Then,
Question: Is there any stable element $f(X,S) \in {\Bbb F}_2[[X,S]]$ by $\sigma$ such that $S \nmid f(X,S)$ ? I.e., $f(\sigma(X),\sigma(S)) = f(X,S) \times \delta(X,S)$ with some non-zero $\delta(X,S) \in {\Bbb F}_2[[X,S]]$ holds.
The motivation is to give alternative proof of Fermat's Last Theorem.
More concretely, we can give different proof of $R = T$ proved by Andrew Wiles. My approach involves a deep analysis on the action $\sigma$ on a certain complete local ring ${\Bbb F}_2[[X,S]]/f(X,S)$. This is the reason why I would like to know that on earth there will be such stable element (f) under the action by $\sigma$.
If I can see that there is NO such element $f(X,S)$ at all even under a more general condition for $\sigma$ as $\sigma \colon X \mapsto X + S + p(X,S)$ with an arbitrary element $p(X,S)$ s.t. ${\mathrm{deg}}\,p(X,S) \geq 2$ and $S \nmid p(X,S)$, this will show a certain Iwasawa algebra will be a power series ring, which can be seen a natural version of Wiles' power series ring gotten by patching argument.