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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.

Arman Malekzadeh
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  • The only correction I would make is $f$ cannot be one-to-one because we have $|F|>|F'|$. I don't know what you mean by "$f$ fails to become a mapping." – M10687 Jun 09 '16 at 16:11
  • There's also the question of whether the answer is one or none; some (all?) people would say that a field homomorphism has to send $1$ to $1$... – David C. Ullrich Jun 09 '16 at 16:14
  • I want to mean that if f is one to one then there are five elements left in F which map nowhere in F' under the mapping f which violates the defination of such a mapping f.Isn't it? –  Jun 09 '16 at 16:15

1 Answers1

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Hint: there does not exist any morphism since the characteristics of the fields are different.

Tsemo Aristide
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