Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

We define the Frobenius Homomorphism as that:

Let $F$ be a field of characteristic $p\gt 0$. Then we call the Frobenius homomorphism this map: $$\phi:F\to F, \phi(x)=x^p$$

I have the following questions:

  1. When this homomorphism is in fact an endomorphism or an automorphism?
  2. Why $x^p=0$ implies $x=0$ (I think it's true in a finite field $F$, but I don't know why)

I would appreciate it so much if someone could help me with this. I think it's a common doubt for beginners because I've already read some books without any clarification on this topic.

Thanks again

share|cite|improve this question
up vote 1 down vote accepted

First, this homomorphism is always an endomorphism, because endomorphism means homomorphism from a space to itself. Indeed, $\phi$ maps $F$ to itself.

The frobenius map is an automorphism if and only if it is bijective and has a bijective inverse. What this says is that "$\phi$ is an automorphism" is equivalent with "$F$ contains a unique $p^{\mathrm th}$ root for every element $x\in F.$" In other words, $F$ is a perfect field of characteristic $p.$

Second, the fact that $x^p=0\Rightarrow x=0,$ is true because $F$ is a field, and thus has nilradical $(0).$ (Suppose $x\neq 0.$ Can you derive a contradiction?)

share|cite|improve this answer
In the second question, can I say also because $F$ is in particular an integral domain? Thank you for your answer :) – user42912 Dec 12 '12 at 0:31
Dear @RafaelChavez, yes that certainly works. Or, since $F$ is a field, if we suppose $x\neq 0,$ then $x$ is invertible, and we cannot have $x^n=0.$ You're welcome! – Andrew Dec 12 '12 at 0:34
Andrew, I didn't understand why you said that: "this homomorphism is always an endomorphism, because endomorphism means homomorphism from a space to itself". Endomorphisms are surjective homomorphisms and we have counterexamples of homomorphism from a space to itself, without be endomorphisms. – user42912 Dec 15 '12 at 7:29
Dear @RafaelChavez, no endomorphisms need not be surjective. – Andrew Dec 15 '12 at 16:29
yes, you're right, thank you – user42912 Dec 16 '12 at 8:49

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.