Show that there exists a finite field with $p^{2}$ elements for every prime $p\in\mathbb{N}$.
What I thought is that if I find some irreducible polynomial of degree two over $\mathbb{Z}_p[x]$, then I will be done. But I had find it for some particular prime $2,3,5.$ But how to find irreducible polynomial of degree two in general in $\mathbb{Z_p[x]}$?
Best Answer
Let $p=2$. Note that $x^2+x+1$ is irreducible over the two-element field, for it has no roots in that field.
Now let $p$ be odd. Let $\mathbb F_p$ be the $p$-element field. We have $(-a)^2=a^2$, and if $a\ne 0$ then $-a\ne a$. Thus there are at most $\frac{p-1}{2}$ non-zero elements of $\mathbb F_p$ that are squares of elements of $\mathbb F_p$. (Actually, there are exactly $\frac{p-1}{2}$, but we don't need that.)
So there are at least $\frac{p-1}{2}$ elements of $\mathbb F_p$ that are not the square of any element of $\mathbb F_p$. Let $b$ be a non-square in $\mathbb F_p$. Then $x^2-b$ is irreducible over $\mathbb F_p$.