discrete mathematics - How to construct minimal polynomial?

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 $$ Z_3 [x]/(x^4+x^2+x+1)$$

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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.