Find the minimal polynomial for the roots $\sqrt{2}$ and $\sqrt{3}$ over $\mathbb{Q}$.
I know the definition of a minimal polynomial is defined for a single root, but can we talk about minimal polynomials for multiple roots simultaneously?
I have only seen methods for finding the minimal polynomial of a single root.
Would it just be the product of the minimal polynomials for each root separately?
I am trying to better my understanding of finding the degree of field extensions such as $ \left[ \mathbb{Q}(\sqrt{2}, \sqrt{3}) : \mathbb{Q} \right] $