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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 ${\Bbb Q}$?

David Feldman
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  • Utterly trivial remarks (which I make only because they could have been included in the question): any quadratic poly is already of the form $x^2+ax+b$, and "completing the cube" says that any cubic is of the form $x^3+ax+b$ after a linear change of variables, so the smallest degree of such a $p$, if one exists, is at least 4. – Kevin Buzzard Dec 09 '10 at 10:47
  • Welcome, David! – Jon Bannon Dec 09 '10 at 13:07
  • A stupid remark on your question: you are asking whether all extensions are contained in an extension of Q generated by a root of $x^n+ax+b$. Say you are more ambitious and ask if equality holds, i.e. if it has a generator of this shape. If so, in particular any extension of Q_p has a generator of this kind and any finite extension as well. If n=6 and p=7, all such polynomials are reducible, so this ambitious generality is not true. But you can ask the original question in finite fields to have a hint of what the answer is... – A. Pacetti Dec 09 '10 at 14:52
  • I realized in the cold light of morning (but haven't had the time till now to write), that I simply hadn't asked the question I'd intended.

    I'd simply edit my original question, but now it seems my accidental question has some interest in itself.

    What I had meant to say was:

    Let $L$ stand for the smallest extension of ${\Bbb Q}$ closed under the operation of adjoining all roots of polynomials of the form $x^n+ax+b, a,b∈L$.

    Should I start a new question?

     – David Feldman Dec 09 '10 at 19:37
  • Sure, start a new question, and link to it from here. – Scott Morrison Dec 09 '10 at 22:51
  • Thanks Scott...and done

    http://mathoverflow.net/questions/48855/galois-theory-generalization-of-abels-theorem-better-version

     – David Feldman Dec 09 '10 at 23:59
  • @David: You did not specify whether n is fixed or not. @Kevin: using Tschirnhausen transformations you can reduce, if I recall it correctly, even quartic and quintic polynomials to this special form. A good place to look for confirmation is King's "Beyond the quartic". – Franz Lemmermeyer Dec 10 '10 at 12:22

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