What does $\mathbb{Q}(\sqrt{2},\sqrt{3})$ mean? What set is $\mathbb{Q}(\sqrt{2},\sqrt{3})$? Is it the set $X = \{a\sqrt{2}+b\sqrt{3}:a,b\in\mathbb{Q}\}$?
$\mathbb{Q}(\sqrt{2},\sqrt{3})$ is the smallest field containing $\mathbb{Q}$, $\sqrt{2}$ and $\sqrt{3}$. One can say that $\mathbb{Q}(\sqrt{2},\sqrt{3})$ is the field generated by $\{\sqrt{2},\sqrt{3}\}$ over $\mathbb{Q}$. It an important fact that one has the following tower of extensions: $$\mathbb{Q}\hookrightarrow\mathbb{Q}(\sqrt{2})\hookrightarrow\mathbb{Q}(\sqrt{2},\sqrt{3}).$$
It is false to say that: $$\mathbb{Q}(\sqrt{2},\sqrt{3})=\mathbb{Q}\sqrt{2}+\mathbb{Q}\sqrt{3}.$$
The latest set is not a field. Actually, one has: $$\mathbb{Q}(\sqrt{2},\sqrt{3})=\mathbb{Q}+\mathbb{Q}\sqrt{2}+\mathbb{Q}\sqrt{3}+\mathbb{Q}\sqrt{6}.$$
Not quite.  It is $\{a\sqrt{2}+ b\sqrt{3}+ c : a,b,c \in Q\}$.
$Q(\sqrt{2}, \sqrt{3})$ is "the smallest field containing all rational numbers as well as $\sqrt{2}$, and $\sqrt{3}$" so must contain, for example, 1.