abstract-algebrafield-theoryfinite-fieldsproof-explanation
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?
Perhaps A Book of Abstract Algebra by Charles Pinter? It's very cheap, and if I remember correctly, provides motivation for the theorems/corollaries/etc. It's a small book, and there's a lot of stuff it doesn't cover, but I think it would prepare the reader well for more advanced treatments of algebra.
How about Adventures in Group Theory by David Joyner?
He outlines how you might set up the problems of describing the configuration space of the Rubik's cube and other puzzles. However, I suspect more thorough discussions might be open.
Wikipedia has the Rubik's cube group as $ G=(\mathbb{Z}_2^7\times \mathbb{Z}_3^{11})\ltimes ((A_8 \times A_{12})\ltimes \mathbb{Z}_2)$. Can you think of an element of order $3$ in that group? All the twists, $\{F,B,U,L,D,R\}$ have order $4$. Perhaps there's a sequence where when you repreat it three times, preserves the identity?
Look at any speed-cubing page. The algorithms are slow much less than God's number of 20. What local features could allow you to speed up?
Rubiks.com's own solution page. Can you build your own sequence that preserves the top row? Top two rows? Corner swapping?
These kinds of questions may still have not been attacked theoretically.
Best Answer
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.