> [!NOTE] Theorem (Field has no Proper Zero Divisors) > Let $(F,+,\times)$ be a [[Field (Algebra)|field]]. Then $(F,+,\times)$ has no [[Proper Zero Divisor|proper zero divisors]]: $\forall a,b\in F: ab = 0 \implies a = 0 \lor b =0$ > >Equivalently, $F$ is an [[Integral Domain|integral domain]]. *Proof*. Suppose $a \neq 0$ and $ab=0.$ Then since $F$ is a field, $a$ has a multiplicative inverse $a^{-1}\in F.$ Then $1=a^{-1}a$ and associativity of ring product: $b=a^{-1}ab = 0$by [[Ring Product With Zero is Zero|ring product with zero]]. *Proof*. Follows from [[Division Ring has no Proper Divisors]].