The minimal polynomial of $\sqrt{2}$ is $x^2 - 2$ and the minimal polynomial of $\sqrt{5}$ is $x^2 - 5$. Since these polynomials are irreducible over $\mathbb{Q}$, the extension $E = \mathbb{Q}(\sqrt{2}, \sqrt{5})$ is a Galois extension.
Now, we want to find the
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