In the textbook Linear Algebra by Friedberg/Insel/Spence, it says that the uniqueness of the multiplicative inverse in a field is a consequence of Theorem C.1 (Cancellation Laws).
But my proof does not use cancellation:
Suppose that $1'$ in $F$ satisfies $1'a = a$ for all a in $F$.
Choose $a = 1$. Then $1' \cdot 1 = 1$.
By (F3), $1a = a$ for all $a$ in $F$. Choose $a = 1'$. Then $1 \cdot
> 1' = 1'$.By (F1), it follows that $1' = 1 \cdot 1' = 1' \cdot 1 = 1$. So the multiplicative identity is unique.
So how do I prove that the multiplicative identity is unique using Theorem C.1 (Cancellation Laws)?
Best Answer
Your original proof is perfectly valid. But if you insist on using a cancellation law to prove that if $1,1'$ are both multiplicative identities then $1=1'$, just write $1\times 1=1=1'\times 1$, and then cancel the $1$ from the right to obtain $1=1'$. Contrary to your original proof this does use $1\neq0$; this is an axiom of fields.