Compute the degree of over , where is or .
Since , the degree of over is at most 2. We can solve the linear system in (as a vector space over ) to find a polynomial satisfied by ; evidently is a root of . (WolframAlpha agrees.) Evidently, (WolframAlpha agrees), which is irreducible by Eisenstein’s criterion; so is irreducible. Thus is the minimal polynomial of over , and so the degree of over is 2.
In a similar fashion, , as an element of , has degree at most 3 over . Evidently, is a root of (WolframAlpha agrees). Evidently, , which is irreducible by Eisenstein. So is the minimal polynomial of , and thus has degree 3 over .