Finding a splitting field over Q

[PsychonautQQ]

Homework Statement

I need to construct a splitting field of f(x)=x^4-x^3-5x+5 over Q

Homework Equations

The Attempt at a Solution

So first I will assume r is a root and divide f(x) by (x-r). The quotient came out to be x^3 + (r-1)x^2 + (r^2-r)x + r^3 - r^2 - 5. I am a bit confused what to do now, do i assume another root, call it m perhaps, and do the division again?

 

[pasmith]

Always check first for obvious integer roots.

Most obvious is: 0 is a root if and only if there is no constant term.

Next most obvious is: 1 is a root if and only if the sum of the coefficients is zero.

 

[PsychonautQQ]

Wow, I didn't know that condition for 1 being a root, thanks! I have now reduced my expression to (x-1)(x^3-5). Will the roots of x^3-5 be the cube root of 5, and then the product of the cube root of 5 with the first and second powers of a third root of unity?