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What do elements in $\frac{\mathbb{F}_9}{\mathbb{Z}_3}$ look like?

I construct $\mathbb{F}_9=\frac{\mathbb{Z}_3[x]}{\langle x^2-2 \rangle}$ and now my homework wants me to find the possible minimal polynomials for $z$ over $\mathbb{Z}_3$ where $z \in \mathbb{F}_9/\mathbb{Z}_3$. I want to see what $\frac{\mathbb{F}_9}{\mathbb{Z}_3}$ looks like.

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    "minimal polynomials for $z$ over $\mathbb{Z}_3$" does not mean "$z \in \mathbb{F}_9/\mathbb{Z}_3$". – 2017-02-06
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    It's like "given $z \in \mathbb{F}_9/\mathbb{Z}_3$, determine the minimal polynomials for $z$ over $\mathbb{Z}_3$". – 2017-02-06
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    In which sense are we taking the quotient here? Is this a quotient group? – 2017-02-06
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    I think $\mathbb{F}_9/\mathbb{Z}_3$ means *field extension*, not a *quotient*. – 2017-02-06

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Assuming $\mathbb{F}_9/\mathbb{Z}_3$ means field extension, not a quotient, the question is equivalent to

Find all monic quadratic polynomials that are irreducible over $\mathbb{Z}_3$.

There aren't many candidates.

Another route is to notice that all elements of $\mathbb{F}_9$ are roots of $x^9-x$ and then factor $x^9-x$ over $\mathbb{Z}_3$:

$$ x^9-x= x (x + 1) (x + 2) (x^2 + 1) (x^2 + x + 2) (x^2 + 2 x + 2) \pmod 3 $$

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    I think that's what the problem says. Previously I understood it as quotient and got confused. – 2017-02-06