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**marianne** So, I have to determine the Galois group of $\displaystyle x^4-5$ over $\displaystyle \mathbb{Q}$, $\displaystyle \mathbb{Q}[\sqrt{5}]$, $\displaystyle \mathbb{Q}[\sqrt{-5}]$.

$\displaystyle f(x)=x^4-5$

That is irreducible over $\displaystyle \mathbb{Q}$ by Esienstein's criterion.

Resolvent is:$\displaystyle g(x)=x^3+20x=x(x^2+20)=x(x-\sqrt{-20})(x+\sqrt{-20})$

$\displaystyle \alpha=0, \beta=\sqrt{-20}, \gamma=-\sqrt{-20}$

$\displaystyle

\mathbb{Q}(\alpha, \beta, \gamma)=\mathbb{Q}(0, \sqrt{-20}, -\sqrt{-20})=\mathbb{Q}(\sqrt{-20})=\mathbb{Q}[\sqrt{-5}]$

$\displaystyle

[\mathbb{Q}(\alpha, \beta, \gamma):\mathbb{Q}]=2$

So $\displaystyle G$ is isomorphic to $\displaystyle D_4$ or $\displaystyle \mathbb{Z}_4$.

To determine which, we check is $\displaystyle f$ irreducible over $\displaystyle \mathbb{Q}[\sqrt{-5}]$

$\displaystyle

f(x)=(x^2-\sqrt{5})(x^2+\sqrt{5})$

Is this irreducible or not?

And what to do with $\displaystyle \mathbb{Q}[\sqrt{5}]$, $\displaystyle \mathbb{Q}[\sqrt{-5}]$?