# Multiplicative Identity is Unique

## Theorem

Let $\struct {F, +, \times}$ be a field.

Then the multiplicative identity $1_F$ of $F$ is unique.

## Proof

From the definition of multiplicative identity, $1_F$ is the identity element of the multiplicative group $\struct {F^*, \times}$.

The result follows from Identity of Group is Unique.

$\blacksquare$