This is an exam question from last semester.
We have the finite field $ \mathbb F_{81} = \mathbb Z_3 [x]/(x^4+x^2+x+1)$
(a) Prove that the polynomial $ x^4+x^2+x+1 $ is irreducible
(b) Construct the minimal polynomial of the element $ x^3+x^2+x+1 \space\epsilon\space Z_3 [x]/(x^4+x^2+x+1)$
Use y as a formal variable in this polynomial. Hint: using $ x^3+x^2+x+1 = (x^2+1)(x+1) $ should help with the calculations. (c) Construct the subfield F9 in $ Z_3 [x]/(x^4+x^2+x+1)$
I tried a and I think you can prove it by showing the polynomial has no Zeros? So assuming we call the polynomial g(x). I just filled in {0,1,2} and none of them gave 0 --> You can't split up the polynomial in polynomials of lower orders -> it's irreducible?
I don't know how to do b and c though. Can someone please tell me how to do it in general and what the solution is here? Really need the answer.