General solutions for algebraic equations with fractional degrees

[cryptist]

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 $$x^{2.5}$$ ?

 

[SteamKing]

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 $$x^{2.5}$$ ?

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.

 

[HallsofIvy]

Of course, you could always define $u= x^{1/2}$ so that $x^{5/2}$ becomes $u^5$. 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.