TITLE: Galois theory: Generalization of Abel’s Theorem?  (Better version!)
QUESTION [15 upvotes]: (Unintentionally I have previously asked a similar and perhaps in itself not uninteresting
question
Galois theory: Generalization of Abel's Theorem? 
but this is what I originally had in mind.)

Let $L$ stand for the smallest extension of ${\Bbb Q}$ closed under the operation of adjoining all roots of all polynomials of the form $x^n+ax+b,a,b∈L$. 
What polynomials $p$ don't split over $L$?  In particular, how low
can one make the degree of such a $p$?  (This 
http://en.wikipedia.org/wiki/Bring%E2%80%93Jerrard_form#Bring.E2.80.93Jerrard_normal_form 
would seem to guarantee degree($p$) $> 5$.)
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 ${\Bbb Q}$?

REPLY [5 votes]: A minor note: it suffices to limit oneself to closure under roots of polynomials of form $x^n + b$ or $x^n + x + b$, since any other polynomial of form $x^n + ax + b$ can be transformed into the latter by the change of variables $x = a^{1/(n-1)}y$ (and $a^{1/(n-1)}$ is 'available' by virtue of the former).