I am trying to solve the following problem, Find the degree of the splitting field of the polynomial $p(x)=x^5-3x^3+x^2-3$ over $\mathbb{Q}.$
My approach for solution: Clearly -1 is a root of the given polynomial, so $p(x)$. So, $p(x)=(x+1)\underbrace{(x^4-x^3-2x^2+3x-3)}_{g(x)}$. Now I have to find out the roots of $g(x).$ But I am unable to find the roots. Although I have calculated in mathematica and got all the roots, but how to do it manually?
On
You can factor further this polynomial: $$p(x)=(x^2-3)(x^3+1)=(x^2-3)(x+1)(x^2-x+1),$$ hence its splitting field is that of $q(x)=(x^2-3)(x^2-x+1)$. Observe $x^2-x+1$ remains irreducible in $\mathbf Q(\sqrt 3)$,as its roots are not real ($\mathrm e^{\pm\mathrm i\tfrac\pi3}$).
So the splitting field is $\,\mathbf Q(\sqrt 3,\mathrm e^{\mathrm i\tfrac\pi3})$ has degree $4$ over $\mathbf Q$.
The original equation clearly has $x^2-3$ as a factor to me (actually I noticed that before noticing $x+1$). factoring that it gives $(x^2-3)(x^3+1)$. As you noted, we can factor out the $x+1$ to get $(x+1)(x^2-3)(x^2-x+1)$ which is fully factored, since $x^2-x+1$ is a cyclotomic polynomial and $3$ is not a perfect square. Can you take it from here?