I understand what an ideal is! Since $\mathbb{Q}(\sqrt{2}, \sqrt{11})$ is a field, it has two ideals, {0} and itself.
Non-trivial ideal of $\mathbb{Q}(\sqrt{2}, \sqrt{11})$ is $\mathbb{Q}(\sqrt{2}, \sqrt{11})$ itself.
Therefore, $\mathbb{Q}(\sqrt{2}, \sqrt{11})$ divided by its non trivial ideal is isomorphic to singleton set {0 + $\mathbb{Q}(\sqrt{2}, \sqrt{11})$}.
However the answer was simply singleton {0}. Why was my answer wrong?