Find all irreducible polynomials of degree $2$ over $\mathbb{Z}_5$ Obviously if I write all the possible ones and try the roots I'd get a LOT of polynomials $(125)$ and I'd have to test $5$ roots for each of them, which would be a LOT. Is there any idea?
I must also do it for degree $\le 3$ over $\mathbb{Z}_3$
Do you guys have any ideas to make it easier?
please remember that I'm on a ring theory course 

You can use this proposition: The polynomial $x^{p^{n}}-x$ is precisely the product of all the distinct irreducible polynomials in $\mathbb{F}_{p}[x]$ of degree $d$ where $d$ runs through all divisors of $n$. Then $x^{25}-x$ is the product of all irreducible polynomials over $\mathbb{F}_{5}$ of degree $1$ and $2$. The degree $1$ one's are: $x, x+1,\cdots, x+4$. Factor $x^{25}-x$ into irreducible polynomials and collect the degree $2$ terms.
\begin{equation}
\begin{split}
&x^{25}-x
=x(x^{24}-1)
=x(x^{12}+1)(x^{12}-1)
=x(x^{12}-4)(x^{12}-1)\\
&=x(x^{6}+2)(x^{6}-2)(x^{6}+1)(x^{6}-1)\\
&=x(x^{6}-2^{3})(x^{6}+2^{3})(x^{6}-4)(x^{6}-1)\\
&=x(x^{2}-2)(x^{4}+2x^{2}-1)(x^{2}+2)(x^{4}-2x^{2}-1)\\
&\quad(x^{3}-2^{3})(x^{3}+2^{3})(x^{3}-1)(x^{3}+1)\\
&=x(x^{2}-2)(x^{4}+2x^{2}-1)(x^{2}+2)(x^{4}-2x^{2}-1)\\
&\quad(x-2)(x^{2}+2x-1)(x+2)(x^{2}-2x-1)\\&\quad(x-1)(x^{2}+x+1)(x+1)(x^{2}-x+1)\\
\end{split}
\end{equation}
We now factor $x^{4}+2x^{2}-1$. Let $x^{4}+2x^{2}-1=(x^{2}+ax+b)(x^{2}+cx+d)$. Compare $x^{3}$ term, $c=-a$. $a\neq 0$, otherwise $b\neq 1,-1$, since $x^{2}+1$, $x^{2}-1$ reduce. $b\neq 2,-2$ since $x^{2}+2$ and $x^{2}-2$ are already in the list. Compare $x$ term, $ad=ab$, $d=b$. So $x^{4}+2x^{2}-1=(x^{2}+ax+b)(x^{2}-ax+b)$. $b^{2}=-1$, $b=2,-2$. If $b=2$, compare $x^{2}$ term, $-a^{2}-1=2$, $a^{2}=2$. No such $a$ exists. So $b=-2$. $-a^{2}=1$. $a=2, -2$. Both cases gives: $x^{4}+2x^{2}-1=(x^{2}+2x-2)(x^{2}-2x-2)$.
Now, let $x^{4}-2x^{2}-1=(x^{2}+ax+b)(x^{2}-ax+b)$. $b=2, -2$. If $b=-2$, $-a^{2}+1=-2$, $a^{2}=-2$, impossible. $b=2$, $a=1$. $x^{4}-2x^{2}-1=(x^{2}+x-2)(x^{2}-x-2)$. 
All the degree $2$ irreducible polynomials are: 
\begin{equation}
x^{2}-2,x^{2}+2,x^{2}+2x-1,x^{2}-2x-1,x^{2}+x+1,x^{2}-x+1,x^{2}+2x-2,x^{2}-2x-2,x^{2}+x-2,x^{2}-x-2.
\end{equation}
