(a) Let $p$ be a prime and let b be a non zero element of the field $\mathbb{Z}_{p}$. Show that $b^{p-1} = 1$. Hint Lagrange
My attempt
$|\mathbb{Z}_{p} \setminus \{0\}| = p-1$
since $b \in \mathbb{Z}_{p} \setminus \{0\}$
$\langle b\rangle \leq \mathbb{Z}_{p} \setminus \{0\}$
From Lagrange $\langle b\rangle | \mathbb{Z}_{p} \setminus \{0\}$
suppose $\langle b \rangle = n$
then $\implies n \mid p-1$
$\implies p-1 = nk$
$\implies b^{p-1}=(b^{n})^{k} = e^{k} = e = 1$
(b) Use (a) to prove that if $p$ is prime and a is an integer the p divides $a^{p}-a$.
And now I have to use the above fact to prove Fermat's Little Theorem.. any hints?