Which of the following is (are) correct?

Question

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

$(A)$$F=\mathbb Z/3\mathbb Z$.

$(B)$$F=\mathbb Z/2\mathbb Z$ or $Characteristic$ of $F=3$.

$(C)$$F=\mathbb Z/2\mathbb Z$ or $\mathbb Z/3\mathbb Z$.

$(D)$$Characteristic$ of $F$ is $3$.

Solution:

If $F=\mathbb Z/3\mathbb Z=${$0,1,2$} or $F=\mathbb Z/2\mathbb Z=${$0,1$},then elements of $F$ are satisfying the operation preserving properties.This leads me (PLEASE CHECK!!) to the selection of options$(A)$,$(B)$ and $(C)$.

I'm not getting how to accept or discard option $(D)$.

Please give some suggestions about my attempt......

Answer 1

I guess you are to make the following observations (you justify the unclear conclusions):

1. If $f$ is a homomorphism of fields, then $$2=1+1=f(1)+f(1)=f(1+1)=f(2)=2^3.$$ This implies that $F$ has characteristic either two or three.
2. $f$ is the identity mapping of $\Bbb{Z}/2\Bbb{Z}$.
3. If $F$ has characteristic three, then for all $z_1,z_2\in F$ we have $$F(z_1+z_2)=(z_1+z_2)^3=z_1^3+3z_1^2z_2+3z_1z_2^2+z_2^3=z_1^3+z_2^3.$$ This implies that $f$ is a homomorphism of rings.
4. If $F$ has characteristic two, and there exists an element $z\in F$, $z\neq0,1$, then we can conclude that $z+z^2\neq0.$
5. If $F$ and $z$ are as in step 4, then $$f(1+z)=(1+z)^3=1+3z+3z^2+z^3=1+z+z^2+z^3=f(1)+f(z)+(z+z^2),$$ so $f$ is not a homomorphism of rings.
6. Putting all this together we see that (B) is correct and the rest of the claims are false. For full credit it is important that you understand the logic of this conclusion. I gathered that this may be a difficult step.