Field - Imported Knowlpedia Content

A field is a commutative ring with $1\neq 0$ such that every nonzero element is a unit (equivalently, every $a\neq 0$ has a multiplicative inverse).

Remarks

Fields are precisely rings with only the “trivial” ideals: $R$ is a field iff its only ideals are $(0)$ and $R$ , and iff $(0)$ is maximal.

Examples

$\mathbb{Q}$ and $\mathbb{R}$ are fields.
For a prime $p$ , $\mathbb{F}_p=\mathbb{Z}/p\mathbb{Z}$ is a field.
$\mathbb{Z}$ is not a field since $2$ has no inverse in $\mathbb{Z}$ .