Saturday, January 6, 2018

abstract algebra - How to prove the uniqueness



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



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.


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