Field has no Proper Zero Divisors

Theorem
Let $\struct {F, +, \times}$ be a field whose zero is $0_F$ and whose unity is $1_F$.

Then $\struct {F, +, \times}$ has no proper zero divisors.

That is:
 * $a \times b = 0_F \implies a = 0_F \lor b = 0_F$