How many meanings does $\mathbb{Z}_p$ have?

Question

About notations. $\mathbb{Z}_p$ can mean the group $\mathbb{Z}/p\mathbb{Z}$, that is cyclic group of order $p$. Or it can be a finite field. Could everyone share other meanings? Such a list may be helpful for reading text in which notation is not clarified enough.

Answer 1

For every natural number $n$, the set $\mathbb{Z}_n := \{0,1,\dots,n-1\}$.

The pair $(\mathbb{Z}_n,+_n)$ is a cyclic group of order $n$, with $+_n$ denoting addition modulo $n$.

For every natural number $n$, the set $\mathbb{Z}/n\mathbb{Z} := \{ x+n\mathbb{Z} \mid x \in \mathbb{Z}\}$. If $r := x\bmod{n}$, it can be shown via the division algorithm that $r+n\mathbb{Z} = x + \mathbb{Z}$. If $$ (x+n\mathbb{Z}) + (y +n\mathbb{Z}) := (x+y) + n\mathbb{Z}, $$ then $(\mathbb{Z}/n\mathbb{Z},+)$ is a group; moreover, since $|\mathbb{Z}/n\mathbb{Z}|=4$ and $\mathbb{Z}/n\mathbb{Z} = \langle 1+n\mathbb{Z}\rangle$, it follows that $(\mathbb{Z}_n,+_n) \cong (\mathbb{Z}/n\mathbb{Z},+)$.

Thus, it is slightly abusive to use the notation $\mathbb{Z}/n\mathbb{Z}$ for a finite cyclic group.

Furthermore, it should be noted that if $p$ is a prime, then $(\mathbb{Z}_p,+_p,\times_p)$ is a field. So it's slightly abusive to say that "$\mathbb{Z}_p$ is a field but also a group". Two binary operations need to be specified for a field, whereas groups only have one.

However, because this is tedious, many authors drop the operations or abusively use $\mathbb{Z}/n\mathbb{Z}$ to indicate the finite cyclic group containing $n$ elements.

Answer 2

Those two are 1) Basically the same, and 2) relatively easy to tell apart in context (do you need multiplication to exist or not?) So it's not as bad as you seem to think.

A slightly bigger problem is that it could mean the $p$-adic integers, which is an uncountably infinite ring (integral domain for prime $p$, and almost never used for composite $p$).

It could also mean the localisation of the integers at the element $p$, i.e. the ring of $p$-adic fractions, the subring of $\Bbb Q$ with fractions being a power of $p$.

Answer 3

There are only two definitions of $\mathbb{Z}_p$, and which one is usually clear from the context.

1 . $\mathbb{Z}_p$ is the quotient ring $\mathbb{Z}/p\mathbb{Z}$. As a group with respect to addition, it is cyclic of order $p$. As a ring, it is a finite field with $p$ elements. In particular, it is finite.

2 . $\mathbb{Z}_p$ is the inverse limit of rings $\varprojlim\limits_n \mathbb{Z}/p^n\mathbb{Z}$. It is uncountable.