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Let $E=\mathbb{Q}(\sqrt{2},\sqrt{3})$ and $$ L = E \left( \sqrt{ ( \sqrt{2}+2 ) ( \sqrt{3} + 3)} \right) \ . $$ I want to show that $L/\mathbb{Q}$ is a Galois extension with the Quaternion group as its Galois group.

I know $E/\mathbb{Q}$ is Galois and $L/E$ is also Galois, but it is not true in general that if $K_1/K_0$ is Galois and $K_2/K_1$ is Galois then $K_2/K_0$ is Galois (take $K_0 = \mathbb{Q}$, $K_1 = \mathbb{Q}(\sqrt{2})$ and $K_2 = \mathbb{Q}(\sqrt[4]{2})$ as a counter-example).

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