Consider the quotient ring $R=\mathbb{Q}[x]/\langle x^4-1\rangle$. Prove or disprove the following: $R$ is isomorphic to a product of fields.
I'm new to learning algebra, but usually in questions regarding quotients of groups or rings, a standard technique is to consider the ideal in question (or subgroup) and demonstrate it to be the kernel of some surjective homomorphism, and then applying the First Isomorphism Theorem. In the context of rings of polynomials, I have often found useful to use evaluation homomorphisms, but I am stuck here.
One more fact I know is that if $p(x) \in F[x]$, where $F$ is a field, and we have that $p(x)$ has a root lying outside the field, say $\alpha$, then $F(\alpha) \approx F[x]/\langle p(x) \rangle$, where $F(a)$ is the field obtained by adjoining $\alpha$ to $F$. I don't know how this fact is helpful here. The roots of the given polynomial are ${1, -1, i, -i} \in \mathbb{C}$.
Hint: if $F$ is a field and $f(x),g(x)\in F[x]$ are nonzero coprime polynomials, then $$ F[x]/(f(x)g(x))\cong F[x]/(f(x))\times F[x]/(g(x)) $$