Let $F$ be a field with four elements and let $a$ be an element which is not $0$ nor $1$.
I have shown that $F = \{0, 1, a, a^2\}$ is cyclic, and also that $1+a+a^2=0$.
I am trying to show that $f=x^2 + a$ is reducible and $g=x^2 + x + a$ is irreducible in $F[x]$. This is equivalent to $f$ having a root and $g$ having no roots since they both have degree 2.
I can see that $f(0)$, $f(1)$ and $f(a)$ are not zero, but I can't see how to show that $f(a^2) = a+a$ is zero. I also don't know how to reason about e.g. $g(1) = 1 + 1 + a$. It seems the problem is that I don't understand how $+$ works in $F$. How do I know which element $1 + 1$ is?
Any pointers are much appreciated.
This is a field of characteristic $2$ (as it has $2^2$ many elements). So $b + b = (1 + 1)b = 0b = 0$ for all $b$. In particular, $a+a = 1+1 = 0$.