Questions tagged [galois-extensions]

For questions about Galois extensions of fields. We say that an algebraic extension $L/K$ is a Galois extension iff the subfield of $L$ that is fixed by automorphisms of $L$ which fix K is exactly $K$.

A Galois extension is a field extension that is normal and separable. See Wikipedia for more information.

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

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.
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