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General solutions for algebraic equations with fractional degrees

  1. Oct 25, 2014 #1
    From Abel–Ruffini theorem, we know that, there is no general algebraic solution to polynomial equations of degree five or higher. So there are general solutions for degrees n={1,2,3,4}. Does degree have to be an integer? What about the fractional degrees? Are there general solutions for example for $$x^{2.5}$$ ?
     
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  3. Oct 25, 2014 #2

    SteamKing

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    The fundamental theorem of algebra deals only with polynomials having a finite number of non-zero terms, each term consisting of a constant multiplied by a finite number of unknowns raised to whole number powers.

    http://en.wikipedia.org/wiki/Algebra

    Polynomials having fractional or decimal powers AFAIK do not have a general theory for finding solutions.
     
  4. Oct 28, 2014 #3

    HallsofIvy

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    Of course, you could always define [itex]u= x^{1/2}[/itex] so that [itex]x^{5/2}[/itex] becomes [itex]u^5[/itex]. If you have fractional powers of x with a number of different denominators, you can define u to be x to the 1 over the least common multiple of the denominators to convert to a polynomial in u.
     
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