3. (20 pts) Let $F=Z s(\alpha)$, where $\alpha$ is a zero of $x^{2}+x+1$ in the algebraic closure of $Z_{5}$. (a) Show that $x^{2}+x+1$ is irreducible over Zs. (b) Write $(\alpha+2)^{-1}$ into the form $a_{0}+a_{1} \alpha$. (c) Find the zeros of $x^{2}+x+2$ in the form of $a_{0}+a_{1} \alpha$. $\left(d^{*} ight)$ Find the product of all monic irreducible polynomials of degree two over $Z_{5}$.