Z/pZ is a field for p prime

Builds on Field A finite integral domain is a field

Statement

Let $n \ge 2$ be an integer. The ring $\mathbb{Z}/n\mathbb{Z}$ of integers modulo $n$ is a field if and only if $n$ is a prime number.

Remarks

If $n = a b$ with $1 < a, b < n$ , then the classes of $a$ and $b$ are nonzero and their product is $0$ . So $\mathbb{Z}/n\mathbb{Z}$ is not an integral domain. If $n = p$ is prime and $p \nmid a$ , then Bézout's identity gives $a x + p y = 1$ for some integers $x, y$ . So $x$ is an inverse of $a$ modulo $p$ . The field $\mathbb{Z}/p\mathbb{Z}$ is often written $\mathbb{F}_p$ . It is the smallest finite field of characteristic $p$ .

Reading path

To understand this page from the ground up, read these 9 pages in order. The foundations come first.

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Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

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Group concept
Ring definition
Commutative ring definition
Unit of a ring definition
Zero divisor definition
Field definition
Integral domain definition
A finite integral domain is a field statement
Z/pZ is a field for p prime statement

Tag	`0NV6E43P4A`
URI	`m2://0NV6E43P4A`
Kind	statement, theorem
Writing	W0 stub
MSC	11A07
Status	This copy of the site has no build data. Run `m2 status 0NV6E43P4A`.
Uses	Field, A finite integral domain is a field

Z/pZ is a field for p prime

Statement

Remarks

Reading path

Further reading

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