Show that $f(x)=x^2+2x-1 \in \mathbb{Z}_3[x]$ is irreducible over $\mathbb{Z}_3$. Using this fact construct a finite field $\mathbb{F}_9$ of $9$ elements. If $\alpha$ is a root of $f(x)$, then find the elements of this field. Also find the discrete logarithms of $(\alpha-1)$ to the base $\alpha$ and of $\alpha$ to the base $(\alpha-1)$.
My approach:
Since $f(x)$ has order $2$, it is reducible in $\mathbb{Z}_3$ iff it has a root. Since none of $0$ or $1$ or $2$ satisfy $f(x)=0$, so it is irreducible in $\mathbb{Z}_3$. Now a field of 9 elements is given by $\frac{\mathbb{Z}_3[x]}{<x^2+2x-1>}$. I'm not sure if this is the field that the question is asking me to construct. Also, I need help with the later part of the question, is the question asking me to find the elements of this field in terms of $\alpha$? If yes, how to do that ?
Thanks in advance!
Taking a shortcut. In this field we have $$ \alpha(\alpha-1)=\alpha^2-\alpha=1. $$ So $\alpha$ and $\alpha-1$ are each others inverses. Hence the two discrete logarithms are both ____ (you fill in the blank, observing the ring the discrete logarithm takes its values in).