Irreducible polynomial with given discriminant

Question

Let $F$ a field and let $D$ an element of an extension $K/F$ such that $D$ is not the square of any element of $F$. Does there exists an irreducible polynomial in $F[X]$ with discriminant $D$? Are there sufficient conditions for such polynomial to exists e.g. $char F\neq 2$?

Answer 1

If $D\notin F$, then such a polynomial will never exist. The discriminant of a polynomial $f\in F[x]$ is a polynomial function of the coefficients of $f$. Hence, $disc(f)\in K$.