What are the splitting fields for the polynomials
$f(x)=x^4 + 5x^2 +4 $
and
$f(x)=x^4 - x^2 - 2 $
I know that any polynomial has a splitting field and by using the proof of this fact $f(x)$ of degree $n$ with roots $c_1....c_k$ can be written as $f(x)=(x-c_1)...(x-c_k)g(x)$ where $g(x)$ has no roots in F. Then factor g(x) into irreducible polynomials, choose one, and pass to the field where F[x]/ and then keep repeating, but I am having a very hard time actually implementing. I know the general idea but can't seem to put it into practice.
(1) $0=x^4+5x^2+4=(x^2+1)(x^2+4)\Rightarrow x= \pm i, \pm 2i.$ So the splitting field of this polynomial is $\mathbb Q(i).$
(2). $0=x^4-x^2-2=(x^2+1)(x^2-2) \Rightarrow x= \pm i, \pm \sqrt 2.$ So the splitting field of this polynomial is $\mathbb Q(i, \sqrt 2).$
Note: To find the splitting field of a given polynomial over $\mathbb Q,$ first consider it as a polynomial over $\mathbb C.$ Since $\mathbb C$ is algebraically closed, the polynomial will splits into linear factors. Now "add" the roots of the polynomial to $\mathbb Q.$ You can easily show that the resulting field is the splitting field of the given polynomial.