Let $A$ be an integral domain. Prove that if $(x+1)^2=x^2+1$ in $A[x]$, then $A$ must have characteristic $2$.
We have $x^2+2x+1=x^2+1$, so $2=0$, and hence the characteristic must be $1$ or $2$. Now, I don't see anything wrong with the characteristic being $1$. So, is the problem correct?