<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] For (a), a reducible polynomial of degree 4 must have either linear or quadratic factors.  Having linear factors is the same as having a root, and it is easy to plug in $0,1,$ and $2$ and see that they are not roots of the polynomial.  Therefore, we must verify that 
$$x^4+x^2+x+1 \neq (x^2+ax+b)(x^2+cx+d)=x^4+(a+c)x^3+\ldots +bd$$
for any choice of $a,b,c,d\in \mathbb{F}_3$.  If we had equality, then the $x^3$ term tells us $a=-c$ and the constant term tells us $b=1/d$ so either $b=d=1$ or $b=d=-1$.  Fully expanding out, we have
$$ (x^2+ax+b)(x^2-ax+b)=x^4+(2b-a^2)x^2+b^2 $$
which has zero $x$ term, and therefore cannot equal $x^4+x^2+x+1$.
For (b), we need to find the smallest linear relation over $\mathbb{F}_3$ between powers of $y=(x^2+1)(x+1)$ modulo $x^4+x^2+x+1$.  First, we note that because our field is a vector space over the subfield generated by $y$, the minimal polynomial will have degree $1$, $2$, or $4$.  
One can calculate that $y^2\equiv x^3+x^2+x-1\pmod{x^4+x^2+x+1}$ over $\mathbb{F}_3$, and therefore $y^2+2=y$ is the minimal polynomial for $y$.  Presumably the factorization helps for doing the calculation by hand.
For (c), since the minimal polynomial of $y$ is quadratic, the subfield generated by $y$ is isomorphic to $\mathbb{F}_9$.  
</s>