Are field extensions $\mathbb{Q}(\sqrt{2},\sqrt{3})$ and $\mathbb{Q}(\sqrt{2})(\sqrt{3})$ the same thing?

Question

As the title, I came across a question to compute the Galois group for $\operatorname{Gal}(\mathbb{Q}(\sqrt{2},\sqrt{3})/\mathbb{Q}(\sqrt{2}))$ and I'm getting a bit confused about how to approach it.

Answer 1

Note that:

• $\mathbb Q(\sqrt{2})$ is the smallest subfield of $\mathbb R$ that contains $\mathbb Q$ and $\sqrt{2}$. That is, it is the intersection of all subfields of $\mathbb R$ containing $\mathbb Q$ and $\sqrt{2}$.
• $\mathbb Q(\sqrt{2},\sqrt{3})$ is the smallest subfield of $\mathbb R$ that contains $\mathbb Q$, $\sqrt{2}$ and $\sqrt{3}$. That is, it is the intersection of all subfields of $\mathbb R$ containing $\mathbb Q$, $\sqrt{2}$ and $\sqrt{3}$.
• $\mathbb Q(\sqrt{2})(\sqrt{3})$ is the smallest subfield of $\mathbb R$ that contains $\mathbb Q(\sqrt{2})$ and $\sqrt{3}$. That is, it is the intersection of all subfields of $\mathbb R$ containing $\mathbb Q(\sqrt{2})$ and $\sqrt{3}$.

Do you see how the last two items define the same subfield of $\mathbb R$?