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] $
As previous comments point out, you should find a primitive element of the field extension first, in this case $\sqrt2+\sqrt3$ is the element you want to find. Next, it is a standard approach to find such element's minimal polynomial: $$\begin{align} \text{Let } \theta= \sqrt{2}+\sqrt{3}, (\sqrt{2}+\sqrt{3})^2 = 5+2\sqrt{6} &\Rightarrow \theta^2 - 5=2\sqrt{6} \Rightarrow (\theta^2-5)^2=24 \\ &\Rightarrow \theta^4-10\theta^2 + 1= 0 \end{align} $$ It is easy to verify that $f(x)=x^4-10x^2+1$ is irreducible over $\mathbb{Q}$ by the rational root test and solving for a quadratic factor (which doesn't exist). So $f$ is the minimal polynomial for $\theta$.