I want to find roots of $x^2 - 1$ in $\mathbb{C}$ by ideal theory.
I have thought the following:
At first, we examine prime ideals of $\mathbb{C}[x]/(x^2 - 1)$ except $(0)$.
So they are $(x+1)/(x^2 -1)$ and $(x-1)/(x^2 -1)$.
$\mathbb{C}[x]$ is a PID, whence $\mathbb{C}[x]/(x^2 - 1)$ is a PID and hence prime ideals other than $(0)$ is maximal ideal.
By hilbert nullstellensatz maximal ideals corresponds to roots as a maximal ideal $(x-a)$ $\longleftrightarrow$ point a $a$.
Therefore roots of $x^2 - 1$ are $1$ and $-1$.
Is it correct?
Thanks in advance.