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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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A hands-on method is to check that all the conjugates of Now $\sqrt{2}, \sqrt{3}$, and $\sqrt{6} \in \mathbb Q[\theta]$, for example $$ (\theta^2 - 6 -2\sqrt{3} )^2 = 24 + 12\sqrt{3} $$ and it follows that $\sqrt{3} \in \mathbb Q[\theta]$. But now $$ \theta\theta_3 = 2\sqrt{3} $$ so it follows $\theta_3 \in \mathbb Q$ and a similar argument shows that $\theta_1$ and $\theta_2$ are also in $\mathbb Q[\theta]$. So $\mathbb Q[\theta]$ is the splitting field of the minimal polynomial of $\theta$ and as a consequence is Galois over $\mathbb Q$. Now you can check by hand that the automorfisms $\theta\to\pm\theta_i$ behave as expected but I suppose there are much better methods to do this. To do it you should work out the expression of $\theta_i$ in terms of $\theta$, for example: $$\begin{align} \theta_1 &= \frac{-1}{24}(\theta^7 -20\theta^5 + 60\theta^3 + 24\theta) \\\\ \theta_2 &= \frac{1}{12}(\theta^5 -18\theta^3 + 36) \\\\ \theta_3 &= \frac{1}{12}(\theta^7 -22\theta^5 + 102\theta^3-120\theta) \end{align}$$ to work in the field $\mathbb Q[\theta]$ you have to compute with polynomials modulo the minimal polynomial of $\theta$ ($x^8 -24x^6 + 144x^4 -288x^2 +144$).I recommend you to use some computer algebra system to perform these computations. |
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The existing answer is very brute-force and not to my taste. I present an alternate method, essentially copy-pasted from another answer of mine. Let $\sigma$ be the automorphism of $\mathbb Q(\sqrt 2, \sqrt 3)$ over $\mathbb Q$ sending $\sqrt 2$ to $-\sqrt 2$ and fixing $\sqrt 3$. Compute $\sigma(\alpha^2)/\alpha^2$. You will get $(2-\sqrt 2)/(2+\sqrt 2)$. This is $3-2\sqrt 2=(-\sqrt 2 + 1)^2$. So $\sigma(\alpha^2)=\alpha^2(-\sqrt 2 + 1)^2$. If $\alpha$ were in $\mathbb Q(\sqrt 2, \sqrt 3)$, then $(\sigma(\alpha))=\pm (-\sqrt 2+1)\alpha$ and $\sigma(\sigma(\alpha))=\alpha(1+\sqrt 2)(1-\sqrt 2)=-\alpha$, a contradiction as $\sigma$ has order $2$. This proves $\alpha$ has degree $2$ over $\mathbb Q(\sqrt 2, \sqrt 3)$. We now see that $\sigma^4(\alpha)=\alpha$, so $\sigma$ generates a subgroup of order $4$ in the Galois group. Let $\tau$ be the automorphism of $\mathbb Q(\sqrt 2, \sqrt 3)$ over $\mathbb Q$ sending $\sqrt 3$ to $-\sqrt 3$ and fixing $\sqrt 2$. You can do a similar computation with $\tau$. You will find they are both of order $4$ and anti-commute. The only group of order $8$ with such elements is the quaternions, as the only non-commuative groups of order $8$ are $Q$ and $D_8$, and $D_8$ does not have two anti-commuting elements of order $4$. |
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