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Compute width of a $\mathbb Z_3$ extension over $\mathbb Z_3$

Could you please tell me how to solve this problem?
$$\mathbb{Z}_3[X]/(X^2+2X+1)$$ is a field extension of $\mathbb{Z}_3$?

A:

Define the ring $\mathbb Z_3[X]$ to be the set of polynomials with coefficients in $\mathbb Z_3$ and define the ideal $(X^2+2X+1)$ to be the set of polynomials with zero constant term and with coefficients in $\mathbb Z_3$ such that
$$
\sum_{k\ge 0} a_kX^k \in(X^2+2X+1) \iff a_2=0
$$
Now show that $X^2+2X+1$ is prime and that the polynomials with zero constant term are prime ideals.
Note that $(X^2+2X+1)$ is not maximal but the quotient by this ideal is an integral domain.

A:

Just an alternative,
The polynomial $X^2+2X+1$ has to remain irreducible. How to see that? By Eisenstein’s Criterion:
$$x^3+2x^2+2x+1=(x^3+1+x)^2-x(x^2+2x+1)$$
The prime polynomials
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