Compute the splitting field of over , and its degree.
Note that , where and . Evidently, the roots of are , and is a root of if . This yields the (distinct) roots . There are four of these, and so we have completely factored .
The splitting field of over is thus .
Using this previous exercise, we have and .
Evidently, then, the splitting field of is . Since is a root of the irreducible (Eisenstein), this extension of has degree 2.