Find $\gamma$ such that $\mathbb{Q}(\gamma)=\mathbb{Q}(\alpha,\beta)$ where $\alpha$ is a root of $x^3-2x+3$ and $\beta$ is a root of $x^2+x+2$?
General way to determine $\mathbb{Q}(\gamma) = \mathbb{Q}(\alpha,\beta)$ given $\alpha$ and $\beta$