Properties of p-adic numbers for specific p Do the p-adic integers, rationals, or any of its algebraic closures have interesting, unique properties for specific p or subsets of the prime numbers(i.e. the Fibonacci primes or twin primes)? Are there any open problems that have to do with such properties? The p-adic numbers are interesting for many reasons, but this would certainly take it to another level.

Each $p$-adic field $\Bbb Q_p$ has its own peculiar properties. For instance, when $p\equiv1\pmod4$, $\Bbb Q_p$ contains a square root of $-1$. More generally, the group of roots of unity in $\Bbb Q_p$ is of order $p-1$, except when $p=2$. In $\Bbb Q_2$, the roots of unity are just $\{\pm1\}$.
