Let , the with real cube root of 5 and primitive root of 1.
because is irreducible.
But what is ?
because and yet is a subfield of . Also the degree is because has as a root. So the possibilities are 2 or 3.
If it's 2. Then because the min poly has to be monic and
I think that we have to express as a linear combination (with rational coefficients) of and .
I can't see how do to this?
Thanks for any help!