Let $p$ be a prime number and let $\mathbb{Z}_{p}$ denote the set of integers modulo $p$. Let $k$ be an integer with $0 \leqslant k \leqslant p$ and let $A$ be a subset of $\mathbb{Z}_{p}$ of size $k$.

Let $t$ be a non-zero element of $\mathbb{Z}_{p}$. Show that if $a+t \in A$ whenever $a \in A$ then $k=0$ or $k=p$. Deduce that if $1 \leqslant k \leqslant p-1$, then the sets $A, A+1, \ldots, A+p-1$ are all distinct, where $A+t$ denotes the set $\{a+t: a \in A\}$. Deduce from this that $\left(\begin{array}{l}p \\ k\end{array}\right)$ is a multiple of $p$ whenever $1 \leqslant k \leqslant p-1$.

Now prove that $(a+1)^{p}=a^{p}+1$ for any $a \in \mathbb{Z}_{p}$, and use this to prove Fermat's little theorem. Prove further that if $Q(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\ldots+a_{1} x+a_{0}$ is a polynomial in $x$ with coefficients in $\mathbb{Z}_{p}$, then the polynomial $(Q(x))^{p}$ is equal to $a_{n} x^{p n}+a_{n-1} x^{p(n-1)}+\ldots+a_{1} x^{p}+a_{0} .$