Degree of splitting field over $\mathbb Q$

Question

I want to find degree of the splitting field of $x^4 -1$ over $\mathbb Q$.

$\mathbb Q[\omega]$ would be a splitting field as the 4th roots of unity will be $\omega^i , i= 0,1,2,3$.

Now $x^4 -1$ can be written as $(x^2 +1)(x-1)(x+1)$

So what will be $[\mathbb Q(\omega) : \mathbb Q]$ now that the polynomial is reducible over $\mathbb Q[\omega]$ ?

Answer 1

The degree is $2$ since it is generated by $i$ such that $i^2+1=0$.