> [!NOTE] **Definition** (Field) > A *field* $(F, +, \times)$ is a non-[[Zero Ring|zero]] [[Commutative Ring|commutative]] and [[Division Ring|division ring]] (non-zero elements are [[Unit in a Ring|units]]). **Note** that $(F,+)$ and $(F\setminus \{ 0 \},\times)$ are [[Groups|abelian groups]] where $F\setminus \{ 0\}$ is the same as the [[Unit Group of Ring|unit group]] of $F,$ denoted $F^{*}.$ > [!Example] Examples > **Number Fields**: The sets of [[Real numbers|reals]], [[Complex Numbers|complex numbers]] and [[Rational Number|rationals]] are fields under their respective addition and multiplication operations. > > **Finite Fields**: The [[Integers modulo prime is a field and integers modulo composite is not|integers modulo n is a field iff n is prime]]. # Properties Note that [[Subrings of a field is are integral domains|fields are integral domains]]: let $a,b\in F.$ If $ab=0$ then $a=0$ or $b=0.$ Hence they inherit the cancellation property.