There's a natural way to view any SAT instance as a variety: just replace the Boolean algebra $2$ of truth values with the corresponding Boolean ring $\mathbb{Z}/2\mathbb{Z}$. (See my answer to Is there an equivalent concept of a "variety" for SAT?). A SAT instance is solvable iff the corresponding polynomial has a zero.
This raises a natural question: What happens when we extend the field?
Specifically, for a SAT instance $\varphi$, let $p_\varphi$ be the associated polynomial. For a field extension $F\supset \mathbb{Z}/2\mathbb{Z}$, we can ask whether $p_\varphi$ has a point over $F$.
In particular, take "nice" to mean "Galois." Then I'm interested in classifying the "relative unsolvability" of SAT instances. Formally, let $\varphi, \psi$ be SAT instances, presented using $\wedge$ and $XOR$ instead of $\wedge$ and $\vee$ (so that we can think of $\varphi$ and $\psi$ immediately as polynomials). Then say "$\varphi\le_G\psi$" if for every Galois extension $F\supseteq\mathbb{Z}/2\mathbb{Z}$, if $\psi+1$ has a zero over $F$ then $\varphi+1$ has a zero over $F$.
So, for example, $"(x\wedge x)+x"<_G "\perp"$ - the former corresponds to the polynomial $x^2+x+1$, the latter to $1$.
My question is:
What can we say about "$\le_G$?" In particular, how do I tell whether $\varphi\le_G\psi$?
My tags are somewhat speculative - feel free to alter them. In particular the "nonclassical logic" tag is purely a guess on my part.