Exercise 31 (page 16) in Golan - The Linear Algebra a Beginning Graduate Student Ought to Know (3rd Edition):
Let $F$ be a field satisfying the condition that the function $a \mapsto a^2$ is a permutation of $F$. What is the characteristic of $F$?
I am not able to answer, but I managed to do something.
If $p$ is a prime integer and $F = \mathbb{Z} / p \mathbb{Z}$ (so that the characteristic is $p$), I can prove (or at least I think) that
$p = 2 \iff a \mapsto a^2 \text{ is bijective}$
One direction is obvious, while for the other I just note that $[(p-1)^2]_p = [1^2]_p$ so that, if $p>2$, $a \mapsto a^2$ is not injective.
When it comes to more general cases, I would like some hints from you.
If $a\mapsto a^2$ is a permutation, then $1^2\neq(-1)^2$ unless $1=-1$. But, on the other hand, you always have $1^2=(-1)^2$. Therefore, $1=-1$. In other words, the characteristic is $2$.