# Multiplicative Inverse in Field is Unique

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## Theorem

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

Let $a \in F$ such that $a \ne 0_F$.

Then the multiplicative inverse $a^{-1}$ of $a$ is unique.

## Proof 1

From the definition of multiplicative inverse, $a^{-1}$ is the inverse element of the multiplicative group $\left({F^*, \times}\right)$.

The result follows from Inverse in Group is Unique.

$\blacksquare$

## Proof 2

From the definition of a field as a division ring, every element of $F^*$ is a unit.

The result follows from Product Inverse in Ring is Unique.

$\blacksquare$

## Sources

- 1964: Iain T. Adamson:
*Introduction to Field Theory*... (previous) ... (next): $\S 1.2$