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.

$\blacksquare$

Sources

• 1964: Iain T. Adamson: Introduction to Field Theory ... (previous) ... (next): Chapter $\text {I}$: Elementary Definitions: $\S 2$. Elementary Properties
• 1973: C.R.J. Clapham: Introduction to Mathematical Analysis ... (previous) ... (next): Chapter $1$: Axioms for the Real Numbers: $2$. Fields: Theorem $1 \ \text{(ii)}$