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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.

Bernard
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    Yes, those two fields are the same by definition. This is a quadratic extension, hence the galois group is obviously the group with two elements. – Crostul Sep 27 '20 at 09:33
  • They are the same, just like the polynomial rings $K[X][Y]$ and $K[X,Y]$ are the same. – Bernard Sep 27 '20 at 10:02
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    @Bernard I'd argue that $\mathbb Q(\sqrt{2},\sqrt{3})$, $\mathbb Q(\sqrt{2})(\sqrt{3})$ are the same subfields of $\mathbb R$ while $K[X][Y]$ and $K[X,Y]$ are only canonically isomorphic. But that is certainly a little pedantic and off-topic for this question. – Christoph Sep 27 '20 at 10:17
  • @Christoph: I only wanted to explain the intuition behind ‘the same thing’. – Bernard Sep 27 '20 at 10:46

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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 $\mathbb R$?

Christoph
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