Compute the degree of over . Exhibit three distinct subfields of . Deduce that the polynomial found in this previous exercise is irreducible over .
Since we know is a root of , the degree of over is at most 4. In this previous exercise, we showed that contains both and , so that . Now , so we also have . Note that the minimal polynomial of over is (irreducible because is Eisenstein at 2), the minimal polynomial of is (Eisenstein at 2) and the minimal polynomial of is (Eisenstein at 2). So these subfields all have degree 2 over .
Suppose and . Comparing coefficients, we see that and . If , then , a contradiction since . If , then , a contradiction since is not rational. In particular, , so that and are distinct.
Similarly, if , then either or . So and are distinct.
Finally, if , then either or , again a contradiction. So and are distinct.
Now we claim that the degree of over is 2. To see this, note that (since and are distinct) is irreducible over , and so is the minimal polynomial of over . So has degree 2 over ; hence has degree 4 over .
We can summarize this information using the following labeled lattice diagram.