Binomial theorem for prime exponent Could you explain to me why for prime $p$ we have the following?
$$(x+y)^p - (x^p + y^p)= x^p +  \binom{p}{1}x^{p-1}y +  \binom{p}{2}x^{p-2}y^2 +  \binom{p}{p-1}xy^{p-1} + y^p.$$
I found it here: Abstract Algebra - Solutions to Homework (pdf),
solution 5d, page 3.
I would really appreciate your help.
Thank you.

As written, what's in the notes is incorrect. We should have
\begin{align*}
(x + y)^p - (x^p + y^p) &= x^p + \begin{pmatrix} p\\ 1\end{pmatrix}x^{p - 1}y + \begin{pmatrix} p\\ 2\end{pmatrix} x^{p - 2}y^2 + \ldots + \begin{pmatrix} p\\ p - 1\end{pmatrix}x y^{p - 1} + y^p - (x^p + y^p)\\
&= \begin{pmatrix} p\\ 1\end{pmatrix}x^{p - 1}y + \begin{pmatrix} p\\ 2\end{pmatrix} x^{p - 2}y^2 + \ldots + \begin{pmatrix} p\\ p - 1\end{pmatrix}x y^{p - 1}\\
&= \sum_{k = 1}^{p-1}\begin{pmatrix}p\\k\end{pmatrix}x^{p - k} y^k.
\end{align*}
I can only assume the writer of the notes got hungry and left in the middle of writing the solutions, and forgot to write a few terms ($\ldots$) upon returning.
