Let $x = (a,b) \in \mathbb{Q}^2$ and let $p(x,t) = t^2-at+b$. Does there exist an involution $\tau$ of $\mathbb{Q}^2$ such that for all $\tau(x) \neq x$, $x \in \mathbb{Q}^2$ one of the polynomials $p(x,t)$ or $p(\tau(x),t)$ is irreducible in $\mathbb{Q}[t]$. This is not a homework question and I don't know if this question might be considered research level. It is just a question out of curiosity.



A question about involutions and polynomials
Of course, for example $\tau(x)\equiv x$. Well, if you want fixed point free involution, the answer is still yes since there are countably many irreducible polynomials and countably many reducible polynomials, you may match them. Maybe, you need something more about $\tau$, say, require it to be linear?


