Multiplicative Identity is Unique

Theorem
Let $\left({F, +, \times}\right)$ 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 $\left({F^*, \times}\right)$.

The result follows from Identity of Group is Unique.