Proving $Gal(K/\mathbb{Q})$ is $D_4$.

Question

Let $a = \sqrt{2+i}$ and $K$ is the splitting field of minimal polynomial of $a$ over $\mathbb{Q}$. Prove that $Gal(K/\mathbb{Q})$ is $D_4$.

I find the minimal polynomial of $a$ is $p(x)=x^4-4x^2+5$ and its roots are $\sqrt{2+i},-\sqrt{2+i},\sqrt{2-i},-\sqrt{2-i}$. Let $b=\sqrt{2-i}$. So the splitting field of $p$ is $K=\mathbb{Q}(a,b)$. Also by rational root theorem, $p$ is irreducible over $\mathbb{Q}$. Thus $[\mathbb{Q}(a):\mathbb{Q}]=4$. Also $b\not\in\mathbb{Q}(a)$, and minimal polynomial of $b$ over $\mathbb{Q}(a)$ is $x^2-2+i$. Thus $[\mathbb{Q}(a,b):\mathbb{Q}]=[\mathbb{Q}(a,b):\mathbb{Q}(a)]\cdot[\mathbb{Q}(a):\mathbb{Q}]=2\cdot 4=8.$ Now since $K$ is the splitting field over $\mathbb{Q}$ of a separable polynomial, $K/\mathbb{Q}$ is Galois. Hence $|Gal(K/\mathbb{Q})|=8$. Hence the 8 automorphisms are $$a\to \begin{cases}a\\-a\\b\\-b\end{cases}\quad\text{and}\quad b \to \begin{cases}b\\-b\end{cases}.$$ From this how to conclude $Gal(K/\mathbb{Q})$ is isomorphic to $D_4$?

Thanks

Answer 1

This follows from Corollary $4.5$ in K. Conrad's notes. The minimal polynomial is $x^4-4x^2+5$, so its Galois group is $D_4$ of order $8$.

Answer 2

Find the minimal polynomial of $a$. \begin{align*} a &= \sqrt{2+i} \\ a^2 &= 2+i \\ a^2- 2 &= i \\ a^4 - 4a^2 +4&= -1 \\ \end{align*} So $a$ satisfies the polynomial equation $f(x)=x^4-4x^2+5$. You should check that this polynomial is irreducible, say, by applying Eisenstein's criterion to $(x-1)^4-4(x-1)^2 +5 = x^4 - 4 x^3 + 2 x^2 + 4 x + 2 $.

By inspection $b=\sqrt{2-i}$ is another root of $f$. $b$ does not belong to $\Bbb{Q}[\sqrt{2+i}]$, because if it did, $\sqrt{2+i}\cdot\sqrt{2-i}=\sqrt{5}$ would belong to $\Bbb{Q}[i]$ as $a^2$ and $b^2$ do which it does not. So if we call $K$ the splitting field of $f$ we have $[K:\Bbb{Q}]=8$ since it's generated by $a=\sqrt{2+i}$ and $\sqrt{5}$.

For automorphisms of $K/\Bbb{Q}$ we have the obvious choice of complex conjugation, say $\tau: i \rightarrow -i$. If we define $\sigma$ by \begin{align*} \sigma &: \sqrt{2+i} \rightarrow -\sqrt{2-i} \\ \sigma &: -\sqrt{2+i} \rightarrow \sqrt{2-i} \\ \sigma &: \sqrt{2-i} \rightarrow \sqrt{2+i} \\ \sigma &: -\sqrt{2-i} \rightarrow -\sqrt{2+i} \\ \end{align*}

$\sigma$ has order 4 and $\tau \sigma \tau = \sigma ^{-1}$, so the group generated by $\sigma$ and $\tau$ is isomorphic to $D_4$