Hw12 Problem 10

Problem: (+) Let $p$ be prime. Show that in the ring $\mathbb{Z}_p$ we have $(a + b)^p = a^p + b^p$ for all $a,b \in \mathbb{Z}_p$.
Solution: By looking at Pascal's triangle, you can see that the coefficients of any prime-powered polynomial, besides the first and last term, are all multiples of that prime number. Since we are in $\mathbb{Z}_p$, we know that any multiple of $p$ is equal to zero. This cancels out all the inside terms, leaving only the first and last term, $a^p + b^p$.