The question is :
Let $F$ and $F'$ be two finite fields with nine and four elements respectively.How many field homomorphisms are there from $F$ to $F'$?
My effort:
Let us consider a homomorphism $f : F \to F'$. Now since ker $f$ is an ideal of $F$ and $F$ cannot have any non-trivial proper ideal according to the property of field.So, ker $f$ is either $\{0\}$ or $F$.If ker $f$ = $\{0\}$ then $f$ is one to one which is impossible here, since then $f$ fails to become a mapping.Hence ker $f$ = $F$ i.e. $f$ is a trivial homomorphism which is the only possible homomorphism from $F$ to $F'$.Is my work correct at all? Please check it.
Thank you in advance.
Hint: there does not exist any morphism since the characteristics of the fields are different.