If $p$ is prime, what is the difference between $F_p$, $\mathbb{Z}_p$ and $\mathbb{Z}/(p \mathbb{Z})$?

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 $\mathbb{Z}/(p \mathbb{Z})$?

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$F_p$ is very, very mildly ambiguous because it could also refer to the free group on $p$ generators (en.wikipedia.org/wiki/Free_group). Better to use $\mathbb{F}_p$; this also makes it look more like $\mathbb{Q}, \mathbb{R}, \mathbb{C}$, etc. – Qiaochu Yuan Sep 9 '12 at 6:32

1 Answer

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.

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I suppose the superscript in $\mathbb{Z}_p^\times$ should be read "times" which is the operation in the multiplicative group. Is that right? – Sidious Lord Sep 9 '12 at 6:12
That's it, exactly. This is a common convention for the multiplicative group of any field. – Kevin Carlson Sep 9 '12 at 6:18
Or ring! ${}{}{}$ – Qiaochu Yuan Sep 9 '12 at 6:31