Multiplicative Inverse in Field is Unique/Proof 2
Jump to navigation
Jump to search
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
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.
$\blacksquare$