Prove that the polynomial $\displaystyle f(x)=x^2+x+1$ is irreducible over $\displaystyle

\mathbb{Z}_5

$ Hence construct

a field F with 25 elements which extends $\displaystyle \mathbb{Z}_5

$. Does the polynomial $\displaystyle g(x) = x^2+2$ have

a zero in F? In other words, you need to find all the solutions (if any) to the equation

$\displaystyle x^2+2 = 0$

in F.

My attempt.

$\displaystyle f(x)=x^2+x+1$ has no zero thus irreducible over $\displaystyle

\mathbb{Z}_5

$ (treid x=0,1,2,3,4)How do i construct field with 25 elements?

$\displaystyle g(x) = x^2+2$ has no zero in F because this polynomial is also Irreducible over $\displaystyle

\mathbb{Z}_5

$ How do i find solution to $\displaystyle x^2+2 = 0$

in F?

Thank you in advance.