How to prove the uniqueness

Question

I'm trying to solve this question from Fulton's algebraic curves:

I've already easily solved (a) and the existence part of (b). I'm having problems to prove the uniqueness of part (b).

I need help.

Thanks in advance.

Answer 1

If $a\in R$ is invertible and $b\in R$ are such that $a+bt=0$, then $a=b=0$: if $b\ne0$, then $a\in\mathfrak m$, a contradiction.