Fermat's little theorem for p-adic integers

Question

Just a short question:

Is Fermat's little theorem applicable in the p-adic integers $\mathbb Z_p$? If yes, why?

Answer 1

Fermat’s little theorem says that $a^p \equiv a \bmod p\mathbb{Z}$ for $a \in \mathbb{Z}$ and $p$ prime. The same result goes through for $p$-adic integers, since $\mathbb{Z}_p/p\mathbb{Z}_p \cong \mathbb{Z}/p\mathbb{Z}$, so $a^p \equiv a \bmod p\mathbb{Z}_p$ when $a ∈ \mathbb{Z}_p$.

Answer 2

Here’s a neat but very minor application of the general Little Fermat: let $K$ be a finite field-extension of $\Bbb Q_p$. with ring of (local) integers $\mathscr O$ and with the latter having maximal ideal $\mathfrak m$. Then $\mathscr O/\mathfrak m$ will be a finite field with (say) $q=p^f$ elements. Now let $z\in\mathscr O\setminus\mathfrak m$, i.e. a unit element of $K$. If you take the sequence $z,z^q,z^{q^2},\cdots$, where each entry is gotten by raising the previous to the $q$-th power, you’ll have a sequence converging to the $(q-1)$-th root of unity in $\mathscr O$ that’s congruent to $z$ modulo $\mathfrak m$.

An example: $p=q=5$, take $2,2^5,2^{25},2^{125},\cdots$ and the limit will be a square root of $-1$ in $\Bbb Z_5$.