# find total number of maximal ideals in $\mathbb{Q}[x]/\langle x^4-1\rangle$.

find total number of maximal ideals in $\mathbb{Q}[x]/\langle x^4-1\rangle$.

Let $J=\langle x^4-1\rangle$, $R=\mathbb{Q}[x]$. I want to use $(R/J)/(I/J)\simeq R/I$, where $I$ is ideal of $R$ which contain $J$. Then $R/I$ is field, and $R$ is a principal ideal domain. Let $I=\langle f(x) \rangle$ hence $f(x)$ must be irreducible in $R$, so only choice for $f(x)$ are $x-1,x+1,x^2+1$.

So answer should be $3$. Is it right explanation? and better method