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If we have a formula for $0=x^4-3x^2-2x+1$, exactly what characteristic of the equation changes when I add an $x^5$ term to only the RHS (and not the entire equation) so that $0=x^5+x^4-3x^2-2x+1$ cannot be solved in a general algebraic way? Please try to keep the answers at a low level (beginning calculus) if possible. Thanks.

Ovi
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    Maybe for that particular one we can. – Git Gud Jun 27 '13 at 06:32
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    This is not calculus, but algebra and requires Galois theory, the general theory of field extensions and the groups of automorphisms of these extensions, so I'm afraid a request for low level is hard to fulfill ... – Hagen von Eitzen Jun 27 '13 at 06:33
  • I have a feeling what you are asking for can't be done, or at least hasn't been done. The nearest is expositions of the work of Abel, there is a fairly recent book which is nice. Forget the title! – André Nicolas Jun 27 '13 at 06:33
  • @AndréNicolas well maybe if you could provide an answer using whatever type of algebra you need I will read this later when I actually learn it – Ovi Jun 27 '13 at 06:36
  • @Ovi The content of the answer would depend on how much you know. As for the standard theorems which allow you to conclude what you're looking for, those you can find on any Galois theory book. – Git Gud Jun 27 '13 at 06:37
  • The book is by Pesic, "Abel's Proof." Needs less algebraic background than exposition of the work of Galois. But then Abel proved less, nothing about any specific quintic, only about the lack of a "general" formula. There are some nice introductory books on Galois Theory. Books, not MSE answers. – André Nicolas Jun 27 '13 at 06:38

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