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we just had a a few pages about Galois theory and I have trouble understanding it. I want to prove that $K := \mathbb{Q}( \sqrt{5}, \sqrt{−1})$ is a Galois extension of $\mathbb{Q}$ of degree 4 and then I want to determine $Gal(K/\mathbb{Q})$.

I am not sure what I have to show, to prove that it is a Galois extension. Is it:

$$ \mathbb{Q} = \mathbb{Q}( \sqrt{5}, \sqrt{−1})^{G},$$ with $G := Aut_\mathbb{Q}( \mathbb{Q}( \sqrt{5}, \sqrt{−1})) $ ?
If I want to determine $Gal(K/\mathbb{Q})$, do I start by looking at the values of $\sigma(\sqrt{5})$ and $ \sigma(\sqrt{-1})$ ? Which I guess are

$$ : (\sqrt{5}, \sqrt{-1}), (- \sqrt{5}, \sqrt{-1}), ( \sqrt{5}, - \sqrt{-1}), (- \sqrt{5}, - \sqrt{-1})? $$

Can you please help me!

user26857
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Maria
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  • Depends on what criteria you want to use to prove that $K/\mathbb{Q}$ is Galois. Your automorphism are correct by the way. There exist $4$ of them. If you know that if $|\text{Aut}_{\mathbb{Q}}(K)| = [K/\mathbb{Q}:\mathbb{Q}]$ then $K/\mathbb{Q}$ is Galois, you're almost done. – Quotenbanane Jan 21 '22 at 20:00

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