Linked to my recent answer here, I would like to know if linear disjunction is a polynomial condition.
Technically, for each $p,q\in \mathbb{N}_+$, is there a polynomial of two variables $D_{p,q}(x,y)\in \mathbb{Z}[x,y]$ such that, for any algebraic field extension $F\subset \Omega$, and $\alpha,\beta\in \Omega$ with $[F(\alpha),F]=p,\ [F(\beta),F]=q$, one has $$ D_{p,q}(\alpha,\beta)\not=0\Longleftrightarrow [F(\alpha,\beta),F]=pq $$ or - at least - implies ? (with $D\not\equiv 0$)
