$F$ is a field such that for every $a\in F, a^4=a$, then what is the characteristic of $F$?

Question

$F$ is a field such that for every $a\in F, a^4=a$, then what is the characteristic of $F$?

Take any $a,b \in F-\{0\}$. then $(a+b)^4=a+b\implies a+b +4a^3b+6a^2b^2+4ab^3=a+b\implies 4a+4b+6a^2b^2=0$.

Multiplying throughout by $ab$ and using $a^3=b^3=1$, we get $4a^2+4b^2+6=0$. I am not sure how to go from here. Please help. Thanks.

Answer 1

This is possible only when $-1=1$, otherwise $(-1)^4=1\neq-1$. For the example in the field $\mathbb{F}_2$ the identity $a^4=a$ holds.

Answer 2

I think that I figured it out:

We have $(-a) ^4= a^4$ so $-a= a$ for every $a$. It follows that $2a=0$,hence the field is of char $2$.