# Field Unity Divided by Element equals Multiplicative Inverse

## Theorem

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

Let $a \in F$.

Then:

$\dfrac {1_F} a = a^{-1}$

where $\dfrac {1_F} a$ denotes division.

## Proof

 $\ds \dfrac {1_F} a$ $=$ $\ds 1_F \times a^{-1}$ Definition of Division $\ds$ $=$ $\ds a^{-1} \times 1_F$ Field Axiom $\text M2$: Commutativity of Product $\ds$ $=$ $\ds a^{-1}$ Field Axiom $\text M3$: Identity for Product

$\blacksquare$