Can somebody tell me what $\mathbb{Z}_3$ means?
Does it mean the positive integers up to $3$?
$\mathbb{Z}_3\stackrel{?}{=}\{1,2,3\}$.
Or is it something else?
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4In what context? It often means ``the group of integers mod $3$." – Milo Brandt Feb 13 '17 at 14:50
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1Notation rarely means just one thing in all contexts. So can you give us some context? Where did you come across this notation? – Alex Kruckman Feb 13 '17 at 14:50
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1It depends on the context. Usually, it means $\Bbb Z /3 \Bbb Z$, that is, the integers modulo 3. Sometimes, it refers to the 3-adic numbers – Ben Grossmann Feb 13 '17 at 14:51
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2Also might be worth mentioning that "the integers mod 3" are (0, 1, 2) and not (1, 2, 3) – SubSevn Feb 13 '17 at 14:53
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@Sunsevn technically both are correct, with $3$ denoting the additive identity. – Ben Grossmann Feb 13 '17 at 15:00
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And sometimes, it just means the set $\Bbb Z_3={1,2,3}$. – Ben Grossmann Feb 13 '17 at 15:00
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@Omnomnomnom: Really? I don't think I've seen that notation. I've seen $[3]$, but not $\Bbb Z_3$. – tomasz Feb 13 '17 at 15:53
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@tomasz it's rare, but I've seen it on this site. Usually in the context of set theory/combinatorics – Ben Grossmann Feb 13 '17 at 15:57
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What does $\Bbb Z/3$ mean then? – user5280911 Dec 21 '20 at 17:26
1 Answers
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It may mean either the field with $3$ elements $\mathbf Z/3\mathbf Z$ or the ring of $3$-adic integers, i.e. $\;\biggl\{\sum\limits_{k\ge0}a_k 3^k\:\Big \vert\: \forall k,\;0\le a_k\le 2\biggr\}$.
Bernard
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