Multiplicative Inverse in Field is Unique

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 $\struct {F^*, \times}$.

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): 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 $1 \ \text{(iv)}$
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