Irreducibility over $\mathbb{Z}_3$

Question

Suppose $R= \frac{\mathbb{Z}_3[x]}{x^5+2x^4+x^2+2x+1}$, then which of the following is true?

1. The number of ideals in $R$ is 4.
2. $R$ has exactly two prime ideals.
3. R is an integral domain.
4. R[x] is a PID.

I have tried to check if the polynomial $x^5+2x^4+x^2+2x+1$ is reducible. For that, I have checked for roots and then I assumed that the polynomial is reducible and tried to compare the left-hand side and right-hand side of $x^5+2x^4+x^2+2x+1=(mx^3+ax^2+bx+d)(nx^2+ex+f)$ It seems that the polynomial is irreducible. Is there any general way of checking irreducibility in Finite fields? I know there is an Algorithm which is a bit lengthy for this type of problem.

Answer 1

Yes, there is a general way of checking irreducibility of polynomials over finite fields. It is called the Berlekamp algorithm and is based on linear algebra (see your link, which it the Berlekamp–Welch algorithm). For the above polynomial is gives

$$x^5 + 2x^4 + x^2 + 2x + 1=(x^3 + x^2 + 2)(x^2 + x + 2).$$ Your computation is the right way to solve the problem without this algorithm by hand. The only problem seems that, maybe, you did not consider that we have a linear system of equations over $\Bbb F_3$ and not over a field of characteristic zero. So for example $4=1$ for the absolute term.

Answer 2

There are only three irreducible polynomials of degree $2$ in $\mathbb{Z}_3[x]$: $$ x^2 + 1, \quad x^2 + x + 2, \quad x^2 + 2 x + 2 $$ Try to divide your polynomial by each of these.