Field

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).

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 $\mathbb{Z}$.