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(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 ${\Bbb Q}$?

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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).

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