# Definition:Multiplicative Identity

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## Definition

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

The identity element of the multiplicative group $\struct {F^*, \times}$ of $F$ is called the **multiplicative identity** of $F$.

It is often denoted $e_F$ or $1_F$, or, if there is no danger of ambiguity, $e$ or $1$.

## Also known as

The **multiplicative identity** of $F$ can also be referred to as the **unity** of $F$.

This arises from its roles as the unity of the ring that $\struct {F, +, \times}$ is by definition of a field.

The term **unit** is often used for **unity**.

It is preferred that this is not used on $\mathsf{Pr} \infty \mathsf{fWiki}$ as it can be confused with a **unit** of a ring, which is a different thing altogether.

## Also see

- Definition:Unity of Semiring
- Definition:Unity of Ring
- Definition:Field Negative
- Definition:Unit of Algebra

## Sources

- 1944: Emil Artin and Arthur N. Milgram:
*Galois Theory*(2nd ed.) (translated by Arthur N. Milgram) ... (previous) ... (next): $\text I$. Linear Algebra: $\text A$. Fields - 1955: John L. Kelley:
*General Topology*... (previous) ... (next): Chapter $0$: Algebraic Concepts - 1964: Iain T. Adamson:
*Introduction to Field Theory*... (previous) ... (next): Chapter $\text {I}$: Elementary Definitions: $\S 2$. Elementary Properties - 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $3$: Field Theory: Definition and Examples of Field Structure: $\S 87$ - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $0$ Zero - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $0$ Zero