Condition for Division by Field Elements to be Unity

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

Let $a, b \in F$.

Then:
 * $\dfrac a b = 1_F$


 * $a = b$

where $\dfrac a b$ denotes division.

Necessary Condition
Let $a = b$.

Then:

Sufficient Condition
Let $\dfrac a b = 1_F$.

Then: