Artin's Algebra, Chapter 10 problem 5.16 states:
Let $F$ be a field. Prove that the rings $F[x]/(x^2)$ and $F[x]/(x^2-1)$ are isomorphic if and only if $F$ has characteristic 2.
As a pedantic concern: if F has characteristic 0, then surely this isomorphism still holds? So maybe it should be "characteristic at most 2"?
More seriously, it seems like $F[x]/(x^2)=\left\{f_0 + f_1 x\right\}$ since we're just setting $x^2=0$. Similarly, it seems like $f_0 + f_1x / (x^2-1) = f_0 + f_1 x$, which implies that $F[x]/(x^2)=F[x]/(x^2-1)$ independent of the characteristic of $F$. To prove this we need to show that $\text{deg}(fg)=\text{deg}(f)+\text{deg}(g)$, which I believe is true in at least integral domains.
What is my mistake here?