Let $L$ stand for the field obtained by adjoining to ${\Bbb Q}$ all roots of all polynomials of the form $x^n+ax+b$, $a,b\in {\Bbb Q}$.

What polynomials $p$ don't split over $L$? In particular, how low can one make the degree of such a $p$?

Classically, $S_n$ occurs as a Galois group for certain $x^n+ax+b$, $n\geq 5$. That means that obstructions for $p$ splitting over such $L$ must reflect information beyond the Galois group of $p$. So absent a full answer to my question, what candidates does one have for such an obstruction? For example, does the form of the polynomial single out particular representations of $S_n$?

Again, absent a full answer, does the literature contain theorems about polynomials not splitting over similar large extension of ${\Bbb Q}$?