Find the splitting field of $x^4-4x^2+1$ over $\mathbb{Q}$
I proved that the polynomial $x^4-4x^2+1$ is irreducible over $\mathbb{Q}$ , so if $a$ is a root of the polynomial, i got:
$x^4-4x^2+1=(x-a)(x+a)(x^2+a^2-4) $.
I don't know how to prove that the quadratic polynomial is reducible in order to prove that $\mathbb{Q(a)}$ is the splitting field. Thank you all.
It turns out that $\frac 1 a(=4a - a^3)$ is also a root of the polynomial.
Thus it suffices to check that $x^2 + a^2 - 4 = (x - \frac 1 a)(x + \frac 1 a)$, which is easy.