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In the book Contemporary Abstract Algebra by Gallian, I'm reading the following proof of the theorem that every finite field is perfect:

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But what seems questionable to me is the part where the author is arguing that $(a+b)^p= a^p + b^p$. Consider $\mathbb{Z}_3=\mathbb{Z}/3\mathbb{Z}$, which is a field. Then $(a+b)^2=a^2+2ab+b^2$. But does $3$ divide $2$? If so then $2=3n=0\mod 3$, for some integer $n$, which is obviously not true. Would someone please clarify this for me?

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    $\begingroup$ You should take $3$ in your example. This property only holds for the characteristic $p$ of that field, where the characteristic is the smallest natural number $p$ such that $p \cdot 1 = 0$. In your example, the characteristic is $3$ instead of $2$. $\endgroup$ – Student Mar 14 '17 at 18:53
  • $\begingroup$ @RCT why is $\phi$ onto? $\endgroup$ – Babai Jan 31 '18 at 15:22
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At the beginning of the proof, $p$ is defined as the characteristic of the field. The claim is that the Freshman's Dream holds for that power, not just any power.

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  • $\begingroup$ Can you please explain why is $\phi$ onto? $\endgroup$ – Babai Jan 31 '18 at 15:23
  • $\begingroup$ @Babai $\phi$ is a map between finite sets of the same size, so it is surjective if and only if it is injective. $\endgroup$ – RCT Feb 1 '18 at 16:58

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