Additive inverse in a field is unique

Question

there will be an $a$ let's prove $-a$ is unique.

let's assume there are 2 additive inverses $b$ and $c$ therefore $a+b=a+c$

let's multiply them by the multiplicative inverse of $a$

I try to use the multiplicative inverse but it does not work.

Answer 1

Let $b$ and $c$ be two additive inverses for $a$. Then:

$$b=b+0=b+(a+c)=(b+a)+c=0+c=c$$