Field

Builds on Commutative ring Unit of a ring

Definition

A field is a commutative ring $K$ with these two properties.

$1 \neq 0$ in $K$ .
Every nonzero element of $K$ is a unit: for each $a \in K$ with $a \neq 0$ , there is $b \in K$ with $a b = 1$ .

Remarks

The rings $\mathbb{Q}$ , $\mathbb{R}$ and $\mathbb{C}$ are fields. The ring $\mathbb{Z}$ is not a field. For a prime number $p$ , the ring $\mathbb{Z}/p\mathbb{Z}$ is a field. Every field is an integral domain. A division ring has the same axioms without the commutativity of the multiplication.

Reading path

To understand this page from the ground up, read these 5 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
Field definition

Where this leads

195 pages build on this one.

Tag	`0QA8CHH8MG`
URI	`m2://0QA8CHH8MG`
Kind	definition
Writing	W1 stated
MSC	12E99
Defines	Field
Uses	Commutative ring, Unit of a ring

Field

Definition

Remarks

Reading path

Where this leads

Further reading

What this page needs for W2

What links here

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