Compute the splitting field of over , as well as its degree.
Let denote the positive real fourth root of 2. Evidently, . So the splitting field of over is .
Note that has degree 2 over . Over the UFD , is Eisenstein at , hence irreducible, and so by Gauss’ Lemma, is irreducible over (the field of fractions of ). So has degree 4 over . We can visualize this scenario in the following diagram.
So has degree 8 over .