I need to find units and zero divisors in $\mathbb{Q}[x]/{(x^2-1)}$.
I don't know how to start... I know that all elements in $\mathbb{Q[x]}/{(x^2-1)}$ are polynomials $p(x)= ax+b$ like $0, 1, x ,x+1$. Is there any way to find all elements in $\mathbb{Q[x]}/{(x^2-1)}$?
You should also know how addition and multiplication work in $\mathbb{Q[x]}/{(x^2-1)}$: you do them as you would in $\mathbb{Q[x]}$ but remember that $x^2-1=0$.
Two polynomials $f,g$ are going to be zero divisors if $f(x)g(x)=0$. For instance, $(x-1)(x+1)=x^2-1=0$. Can you characterise all of the possible polynomials that equal 0?
Two polynomials $f,g$ are going to be units if $f(x)g(x)=1$. Think similarly to the above, and this time $1=x^2=2x^2-1=\cdots$.