Compute the splitting field of over and its degree.
The roots of exist in (if we don’t know this yet, just assume some roots are in ). Let be such a root; then .
Evidently, . Comparing coefficients, we see that either , , or . If , then , a contradiction in . Likewise, if we get a contradiction. Thus . Substituting, we have , so , and so . There is a unique positive 4th root of , which we denote by ; so and , and hence . There are 4 such roots, and so we have completely factored . (WolframAlpha agrees.)
So the splitting field of is . Note that if and , then , and . Thus .
Note that is a root of , and that the reverse of is . Now is irreducible over the UFD , since it is Eisenstein at the irreducible element . So is irreducible over , the field of fractions of as a consequence of Gauss’ Lemma. As we showed previously, the reverse of , namely , is also irreducible over . So has degree 4 over , and thus has degree 8 over .