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I was working on problems in my math textbook and I saw this problem as a side note and I couldn't figure it out. The author states:

Name as many ways to find the zeros of an equation of nth degree.

I am asking this question because I tried to solve it but I do not understand how to. Can you please help me out?

Shalin Shah
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    Didn't Galois prove this is impossible (I suppose it's an algebraic equation)? – jinawee Jan 07 '14 at 01:17
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    Do you know more or less what they are talking in the book round that question? There are many many methods to find the solutions of an equation of degree $n$. Knowing the topics they are talking about can give methods closer to the techniques in that book. Some possibilities are interval subdivision, or Newton algorithm, or fixed point approximation, for example. –  Jan 07 '14 at 01:19
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    This seems a very vague question. Guess and check? Computer algebra package? Newton's method? Tea leaves? – CPM Jan 07 '14 at 01:20
  • @jinawee That is not what Galois proved. He proved that it is not possible under the extreme restrictions of finding: 1) a general formula, and 2) a formula in radicals. –  Jan 07 '14 at 01:20
  • For once, this might be a legitimate reason for an OP to say "I don't know where to start". This question is so vague and poorly-worded as to not really be a question. – Erick Wong Jan 07 '14 at 02:16
  • @Karene: there is no restriction that require a general formula to be found. The proof was that there are polynomial of degree 5 in which the root cannot be expressed in radicals. In other word, even if you were given one such polynomial and were asked to find the roots of just that polynomial with just radicals, you still cannot do it. The gist of the theorem is more like angle trisection and less like halting problem. – Gina Jan 08 '14 at 03:17
  • @Gina What I was saying to jinawee is that Galois didn't prove that there is no way to solve the problem in the box in Shalin's post (find zeros of equations of n-th degree). The problem in Shalin's post has indeed may ways of solving. As I understand you are telling me what I was telling to jinawee. –  Jan 08 '14 at 03:28
  • @Karene:no, I am making the point against your claim "extreme restriction of finding: 1) a general formula". My point is that the impossibility implied by Galois theorem is not because of the requirement that you need a GENERAL formula. The impossibility come from the failure to express certain root of degree 5 polynomial in radical. Hence my comparison to angle trisection and halting problem. – Gina Jan 08 '14 at 03:38
  • @Karene: in halting problem, the answer are always expressible in "yes" and "no", but impossibility is due to the requirement of a general algorithm that work on all. In angle trisection, the impossibility is because all the points you need to construct a certain specific angle (such as degree 20 angle) cannot be constructed. In halting problem, if you got omnipotent power to foresee the solution and is allowed to give ad hoc solution to each problem, then you can solve it. In angle trisection, you still can't solve it even with such thing being allowed. – Gina Jan 08 '14 at 03:41
  • @Gina Oh, indeed. But what I meant by general solution is that the impossibility doesn't appear for every polynomial of high degree. I see your point. It can give the impression it is negating the impossibility result for particular polynomials. –  Jan 08 '14 at 04:35

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It seems there are formulas (but not in radicals) for general equations of deg $n>4$ using,

  1. Fuchsian functions
  2. Theta functions
  3. Mellin integrals

Notes:

  1. In this MO post (Aug 2010), the OP points out that in Introduction to the Theory of Equations (Conkwright, 1941), on p. 85, the author writes that, "...the general equation of degree n has been solved in terms of Fuchsian functions."
  2. Ottem answers, "... H. Umemura showed that the general algebraic equation can be solved using Riemann theta functions. ".
  3. Carette comments that,"... Apparently, Mellin integrals are enough too."

P.S. There is a similar MO post (Apr 2010), but I believe the answers of the Aug post supersede that of the earlier one.

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