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 $x^2+x+2$?

Question

I have to answer the following problem:

• 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 $x^2+x+2$

I am a bit confused: How do we pick a root of each of those polynomials? If I didn't messed something, I guess they do not have roots in $\mathbb{Q}$ so what roots should we pick? Should we make two extensions so that we obtain a root for each one of them and then try to find $\gamma$?