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 $\struct {F^*, \times}$.
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 $1 \ \text{(iv)}$