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What does the notation $\mathbb Z_n$ mean, where $n$ is also an integer. I have only seen $n$ being a positive integer up to now. some examples are $\mathbb Z_2$ or $\mathbb Z_3$

This is the context: How to prove $x^{2}+x=1$ has a solution in $\mathbb{Z}_{p}$ if and only if $p=5$ or $p\equiv \pm1\bmod 5$

Asinomás
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2 Answers2

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$\mathbb{Z}_n$ is another (shorter) name for $\mathbb{Z}/n\mathbb{Z}$, the ring of residue classes modulo $n$. A residue class modulo $n$ is the set of all integers which give the same remainder when divided by $n$. There are exactly $n$ residue classes, corresponding to the $n$ remainders on division by $n$, $0$ to $n-1$.

The key point is that the remainder of $i+j$ on division by $n$ is the sum of the individual remainders, and the analogue is true for the product. So no matter which member of two given residue classes you choose, the sum will always be in the same residue class.

It only makes sense if $n$ is an integer $\ge 2$. If $n$ happens to be prime, it is a field, that's why those are most interesting. Strictly speaking you could also write $\mathbb{Z}_1$, but that would only contain one element.

VizDracViz
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celtschk
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    And it is somewhat unfortunate when $n$ is a prime, since most people (from what I understand) use $\mathbb{Z}_p$ to indicate the $p$-adic integers and $\mathbb{Z}/(p)$ (or something to this effect) for the usual finite ring. – Andy Jun 02 '13 at 08:50
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    $\Bbb Z_1$ certainly does "make sense" - it is the zero ring ${0}$ where $1 = 0$. Also $\Bbb Z_n$ may denote the isomorphic ring of normal forms ${0,1,2,\ldots,n-1},,$ see my comment on the question. – Bill Dubuque Jan 13 '22 at 19:51
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$p\equiv\pm1$ (mod 5) means (roughly speaking) that the remainder when dividing $p$ by 5 is 1 or -1. So $p$ could be 4,6,9,11,14, etc. $p$ would still be a positive integer though.

john
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