I want to find the splitting field of $f(x)=x^4+2$ over the rationals, $\mathbb{Q}$; and the degree of that splitting field over $\mathbb{Q}$.
So I first solved the equation $x^4=-2$, and get the roots are $x_k=2^{1/4}\exp(i\pi/4)\exp(i\pi\cdot k/2)$. So my splitting field over the rationals is $\mathbb{Q}(2^{1/4}\exp(i\pi/4)\exp(i\pi/2))$; I am not sure how to calculate $[\mathbb{Q}(2^{1/4}\exp(i\pi/4)\exp(i\pi/2)):\mathbb{Q}]$?
Any hints? thanks!
Hint: A solution to this equation is provided by a primitive 8th unit of unity $\xi$, since $\xi^4=-1$. Thus $(2^{1/4}\xi)^4 = -2$. The further solutions refer to $\xi^3,\xi^5,\xi^7$ which are the other primitive 8th roots of unity.