<s>[INST] 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  $$ Z_3 [x]/(x^4+x^2+x+1)$$

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.
 [/INST] I will answer only (b).
By abuse of notation, we write $x$ for the image of $x$ in $\mathbb{Z}_3[x]/(x^4 + x^2 + x + 1)$.
Let $y = x^3 + x^2 + x + 1$.
Since $y = (x^2 + 1)(x + 1)$, $y^2 = (x^2 + 1)^2(x + 1)^2$.
Hence $y^2 = (x^4 + 2x^2 + 1)^2(x + 1)^2 = (-x + x^2)(x + 1)^2 = x(x - 1)(x + 1)^2 = x(x^2 - 1)(x + 1) = x(x^3 + x^2 - x - 1) = x^4 + x^3 -x^2 - x$
$= x^3 - 2x^2 -2x - 1 = x^3 + x^2 + x - 1$.
Hence $y^2 + 2 = y$.
Since $y$ is not contained in $\mathbb{F}_3$, $Y^2 - Y - 1$ is the minimal polynomial of $y$ over $\mathbb{F}_3$.
</s>