Checking if a monic polynomial can be decomposed into linear factors 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?

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. 
