<s>[INST] How many irreductible polynoms of degree $n=3$ are there over $\mathbb Z_{3}=\{\overline{0}, \overline{1}, \overline{2}\}$? How many irreductible polynoms of degree $n=3$ are there over $\mathbb Z_{3}=\{\overline{0}, \overline{1}, \overline{2}\}$?
Please, check my solution:
$f(x)$ is a polynom of degree $n=3 (\delta f=3) \Rightarrow f(x)=a_0 + a_1 x +a_2 x^2 + a_3 x^3$. We have three choices for each coeficient, so we can build $3^4 = 81$ polynoms over $\mathbb Z_{3}$. The reductible polynoms are in the shape $p(x)=(x- \alpha)(b_0 + b_1 x + b_2 x^2)$, so we can build $3.(2^3)=24$ reductible polynoms over $\mathbb Z_{3}$. Conclusion: there are 57 irreductible polynoms.
Thank you!
 [/INST] I assume you mean degree 3, not degree at most 3. We can assume they are monic. If you want the non-monic ones just multiply all these by 2. So I get these irreducible monic polynomials of degree 3 over $F_3$:
$x^3+2x+1$
$x^3+2x+2$
$x^3+x^2+2$
$x^3+x^2+x+2$
$x^3+x^2+2x+1$
$x^3+2x^2+1$
$x^3+2x^2+x+1$
$x^3+2x^2+2x+2$
Being lazy, I just asked Mathematica to print them out, but I seem to have less than you, so maybe I asked in a silly way.
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