# Finding a splitting field over Q

1. Oct 28, 2016

### PsychonautQQ

1. The problem statement, all variables and given/known data
I need to construct a splitting field of f(x)=x^4-x^3-5x+5 over Q

2. Relevant equations

3. 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?

2. Oct 28, 2016

### 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.

Last edited: Oct 28, 2016
3. Oct 30, 2016

### 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?