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