TITLE: If $p$ is prime, what is the difference between $F_p$, $\mathbb{Z}_p$ and $\mathbb{Z}/(p \mathbb{Z})$?
QUESTION [6 upvotes]: It's all in the title: $p$ is prime, what is the difference between $F_p$, $\mathbb{Z}_p$ and $\mathbb{Z}/(p \mathbb{Z})$?  Also, if $p$ is not a prime, what is the difference between $\mathbb{Z}_p$ and $\mathbb{Z}/(p \mathbb{Z})$?

REPLY [3 votes]: These are generally all possible notations for the field of residues modulo $p$. $F_p$ makes it clear that we have the whole field in mind, while $\mathbb{Z}_p$ and $\mathbb{Z}/(p\mathbb{Z})$ can refer to the field, the ring, the module or even just the group under addition. However, none of these notations should properly apply to just the multiplicative group, which can be specified as $\mathbb{Z}_p^\times$.
Of these notations, $\mathbb{Z}_p$ is the least clear, since it can also refer to the $p$-adic integers, or even conceivably $\mathbb{Z}$ localized at $(p)$, though that should really be $\mathbb{Z}_{(p)}$. But in context, it's generally quite clear what's intended.