Let , , and .
- Compute the conjugates of .
- Compute the conjugates of over .
- Find the field polynomial for over , and express it as a power of the minimal polynomial of over .
- Find the conjugates of over and compute the degree of over .
Note that is a root of , which is irreducible by Eisenstein’s criterion. Evidently the roots of (i.e. the conjugates of ) are and .
Certainly , where . Now and are the conjugates of over .
The field polynomial of over is then ; note that is the minimal polynomial of over (it is irreducible by Eisenstein.)
Note that , where . Evidently then the conjugates of over are and . Since has degree 4 over and these conjugates are all distinct, then by Theorem 5.10 in TAN, has degree 4 over .