Finite field such that for every $a \in F$ , $x^2=a$ has a solution for $ x \in F$

Question

Let $F$ be a finite field such that for every $a \in F$, the equation $x^2=a$ has a solution for $x \in F$ , then what can we say about the number of elements in $F$ and characteristic of $F$?

Answer 1

Hint: To say that for all $a\in F$, $x^2 = a$ has a solution in $F$ is to say that the squaring map $x\mapsto x^2$ is surjective. Since $F$ is finite every surjective map $F\to F$ is injective. From this, you should be able to determine the characteristic of $F$...