Suppose you have a finite field $\mathbb{F}$, where $|\mathbb{F}|=p^n$ and $p$ is a prime number. Also suppose that $f(x)=x^2+b\in \mathbb{F}[x]$ is an irreducible polynomial over $\mathbb{F}$ and $r$ is a root of $f(x)$ in an extension of $\mathbb{F}$.
How can I prove that $p\neq 2$?
Let $K$ be a finite field of characteristic $2$. For any $a\in K$, let $f(a)=a^2$. Note that $f$ is one to one. For if $a^2=y^2$, then $(a-y)(a+y)=0$, so $y=a$ or $y=-a=a$.
Since $K$ is finite, and $f$ is one to one, it is onto. Thus everything is a square, and in particular $x^2+b$ is never irreducible.