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I have questions about how to determine if a polynomial can be decomposed into linear factors. If it is not solvable over radicals by Galois Theory, then I am done. But do I have to resort to Galois Theory?

Let the polynomial be:

$$f(x) = x^5 + a x^4 + bx^3 + c x^2 + d x + e $$

where $a,b,c,d$ and $e$ are integers.

I know based on the rational root theorem, I would need to check all factors of “$\pm e$.” However, I do not know the exact values of “$a,b,c,d$ and $e$.” I just know certain properties of them. Also, I cannot use Eisenstein's Criterion since $p^2 \mid e$

Also, I want to use this for higher order monic polynomials with integer coefficients.

Is their a way to answer this in terms of “$a,b,c,d$ and $e$?” Also, based on Galois Theory how can I determine this based on “$a,b,c,d$ $e$” without having to resort to the abstract aspects?

Jesse P Francis
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user152331
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  • I don't think there's anything more you can say beyond the rational root test without more information about the coefficients. – Qiaochu Yuan May 21 '14 at 18:27
  • It sounds to me like you're looking for an analog to the quadratic formula for fifth-order polynomials, and I think no such formula exists. In fact, most polynomials with order 5 or more have no roots that can be expressed in closed-form (see http://en.wikipedia.org/wiki/Algebraic_number#Numbers_defined_by_radicals ) – NovaDenizen May 21 '14 at 18:41

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There are general formulas for the solutions of equations such as:

$ax + b = c$

$ax^2 + bx + c = 0 $

$ax^3 + bx^2 + cx + d = 0$

$ax^4 + bx^3 + cx^2 + dx + e = 0$

In other words, general radical solutions exist up to fourth order solutions. Whether you need to use Galois theory actually depends on the polynomial itself. For example, $x^5 + x^4 - 3x - 3 = 0$ can be solved with radicals since you can just factor it out to $(x^4 - 3)(x + 1)$ and $x^4 - 3$ can be solved with radicals. But what if you have $x^5 + x - 1 = 0$? There is no "easy" way to factor it out. You will actually have to solve it numerically. Whether or not you use Galois theory for $x^5 + x = 1$ or other "irreducible" polynomials, is up to you. Although many of my friends that deal with these types of problems say that Galois theory is the most efficient way.