Solution: We recall the following isomorphism theorem. This ic called the third isomorphism theorem for rings.

We also recall that $R/I$ is a field if and only if $I$ is a maximal ideal in $R$.

In this problem, we note that if $I$ is an ideal of $\left\langle x^4 -1 \right\rangle $, then using the above theorem, we have \[ \frac{\mathbb{Q} [x]/I}{I/\left\langle x^4 - 1 \right\rangle } \cong \mathbb{Q} [x]/I. \] Now $I$ is maximal if and only if $\mathbb{Q} [x]/I$ is a field. Therefore, we need to consider the ideals of $\left\langle x^4 - 1 \right\rangle $ which are irreducible. Note that \[ x^4 - 1 = \left( x^2 + 1 \right) (x-1)(x+1). \] Therefore, ideals of $\left\langle x^4 - 1 \right\rangle $ which are maximal $\mathbb{Q} [x]/\left\langle x^4 - 1 \right\rangle $ will be \[ \left\langle x-1 \right\rangle/\left\langle x^4-1 \right\rangle , \left\langle x+1 \right\rangle/\left\langle x^4-1 \right\rangle \text{ and } (x^2 +1)/\left\langle x^4-1 \right\rangle. \] Hence, there are three maximal ideals of $\mathbb{Q} [x]/\left\langle x^4 - 1 \right\rangle $.