Ring homomorphism and characteristic Let $F$ be a finite field. If $f : F \to F$, given by $ f(x) = x^3$ is a ring homomorphism, then 


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*$ F = \mathbb{Z} / \mathbb{3Z}$

*$ F = \mathbb{Z}/ \mathbb{3Z}$ or  $ F = \mathbb{Z}/  \mathbb{2Z}$

*$ F = \mathbb{Z}/  \mathbb{2Z}$ or characteristic of $F$ is 3.

*Characteristic of $F$ is $3$
For ring homomorphism, $f(x+y)= f(x)+ f(y)$ and $f(xy)= f(x) f(y)$ for $x,y \in F$. So, Charcteristic of $F$ must be 3 so that $ 3x^2y + 3xy^2= 0$, so $4$ option should be true but how to discard option 1?

Consider $(\mathbb Z/3\mathbb Z)[x]/(x^2+1)$. It is a field $\not=\mathbb Z/3\mathbb Z$ of characteristic $3$, so option $(1)$ is not true.

Similarly option $(2)$ is not a necessary condition for $x\mapsto x^3$ to be a ring homomorphism.

For option $(3)$, we shall show that it is necessary for $x\mapsto x^3$ to be a ring homomorphism that either $F=\mathbb Z/2\mathbb Z$ or $F$ has characteristic $3$.
By the binomial expansion, $x\mapsto x^3$ is a ring homomorphism if and only if $3(x^2y+xy^2)=0,\,\forall x, y\in F$. And this is true if and only if $3=0$ or $x^2y+xy^2=0$ in $F$.
The first case means $F$ has characteristic $3$.
The second case says that, when $x,y\not=0$, $xy$ is invertible, and hence $x+y=0$. This means that every non-zero element in $F$ is the additive inverse of $1_F$. As a consequence, there is only one non-zero element in $F$, i.e. $F=\mathbb Z/2\mathbb Z$. 
Thus option $(3)$ is a necessary condition.

Since $\mathbb Z/2\mathbb Z$ has characteristic $2$, option $(4)$ is not necessary.

Hope this helps.
