# FileMaker Pro 18 Advanced V18.0.3.317

Q: Why are there many flags in $\mathbb{C}[x]/(x^{2}-1)$? I need to find the dimension of the quotient ring $\mathbb{C}[x]/(x^{2}-1)$ and since it is easier to work with, I am considering the ring $\mathbb{C}[x]/(x^{2}-1) \cong \mathbb{C}[x]/(x-1) \times \mathbb{C}[x]/(x+1)$. Since it is given that $\mathbb{C}[x]$ is a PID, I am trying to prove that $\mathbb{C}[x]/(x-1)$ and $\mathbb{C}[x]/(x+1)$ are fields and the dimension of $\mathbb{C}[x]/(x-1) \times \mathbb{C}[x]/(x+1)$ is $2$, and therefore the dimension of $\mathbb{C}[x]/(x^{2}-1)$ is $2$. We can see that in $\mathbb{C}[x]/(x-1)$, $x$ is a non-zero divisor (since it is a constant polynomial $1$, but $\mathbb{C}[x]/(x-1)$ is a field) and in $\mathbb{C}[x]/(x+1)$, $x$ is a unit (since it is of degree $1$ and it is a regular polynomial with constant coefficient $1$), so it follows that $(\mathbb{C}[x]/(x-1))^{*} \cong \mathbb{C}[x]/(x+1)$ and $(\mathbb{C}[x]/(x+1))^{*} \cong \mathbb{C}[x]/(x-1)$. Now, why are the fields $(\mathbb{C}[x]/(x-1))^{*}$ and $(\mathbb{C}[x]/(x+1))^{*}$ isomorphic to each other? If $f \in (\mathbb{C}[x]/(x-1))^{*}$, then