Galois group of the polynomial $x^4-2$.

Question

So I'm reading Conrad's notes: https://kconrad.math.uconn.edu/blurbs/galoistheory/galoisaspermgp.pdf and I'm a bit stuck on example 2.3, namely the "effect of $r$ on the roots" part. Having studied the table extensively, I don't see where he gets the fact that say for example, $r(2^{1/4})=2^{1/4}i$.

This is the only thing that is preventing me from grasping the example in full so any clarification on this would be much appreciated.

Answer 1

The table says, rather straight-forwardly, that $r(\sqrt[4]2) = i\sqrt[4]{2}$. That's what the second row of Table 1 shows: the values of $r(\sqrt[4]2), r^2(\sqrt[4]2), r^3(\sqrt[4]2)$, and so on.