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I am trying to explicitly compute the Galois group of $x^4-2$ over $\mathbb{Q}$. I found that the resolvent polynomial is reducible and the order of the Galois group is $8$ using the splitting field $K=\mathbb{Q}(2^{1/4}, i)$. Hence I need to find 8 automorphisms. Thus do I just map elements of the same order to each other that fix the base field $\mathbb{Q}$ $2^{1/4}$ to $i2^{1/4}$

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You can see that $i^n\sqrt[4]{2},\, 0\le n\le 3$ are all the roots, so that

$$K=\Bbb Q(i,\sqrt[4]{2})$$

is the splitting field for the polynomial. Since $L=\Bbb Q(\sqrt[4]{2})$ is real of degree $4$, we see that $K$ is a proper extension of $L$, and since $[\Bbb Q(i):\Bbb Q]=2$ we see the total degree of the extension is $2\cdot 4=8$. But then we have that $\operatorname{Gal}\left(K/\Bbb Q\right)\le S_4$ is a subgroup of $S_4$ of order $8$. This implies it is a Sylow-$2$ subgroup of $S_4$, all of which are isomorphic--by the second Sylow theorem. We know that $D_8$, the dihedral group of order $8$, is such a subgroup, so that gives the isomorphism type.

But then you know what to look for as explicit representations go, you note that relative to the ordering

$$\alpha_j=i^j\sqrt[4]{2}, 1\le j\le 4$$

we have the $4$-cycle $(1234)$ given by the automorphism

$$\begin{cases}\sqrt[4]{2}\mapsto i\sqrt[4]{2} \\ i\mapsto i\end{cases}$$

which is enough to totally determine it, since those are generators of the extension. Clearly also

$$\begin{cases}\sqrt[4]{2}\mapsto \sqrt[4]{2} \\ i\mapsto -i\end{cases}$$

is represented by the transposition $(13)$, and these two generate the group, so give you everything you need for a fully explicit description.

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  • $\begingroup$ I believe the OP is not asking for the isomorphism type, but rather for the actual elements of the group. $\endgroup$ – Matt Samuel Apr 12 '15 at 21:28
  • $\begingroup$ @MattSamuel you may be right, I've edited it to reflect this. Thanks for the comment! $\endgroup$ – Adam Hughes Apr 12 '15 at 21:35
  • $\begingroup$ @AdamHughes: Why do we need to specify where $i$ goes? Wouldn't it be sufficient to know just where the roots go: Say for instance I have $x_1=\sqrt[4]{2}$, $x_2=-\sqrt[4]{2}$, $x_3=i\sqrt[4]{2}$, $x_4=-i\sqrt[4]{2}$. I can choose to map $x_1 \to x_2$ and then $x_1\to x_2 \to x_3 \to x_4$ and get a $2$ and $4-$ cycle respectively. $\endgroup$ – Rutherford Mark Apr 15 '15 at 6:18
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    $\begingroup$ @RutherfordMark that's fine, you're doing the same thing, only with different generators for the field, I chose $i,\sqrt[4]{2}$ because those were the field generators you gave in the original problem. Clearly if you know one you know the other, since $\sigma(i\sqrt[4]{2})=\sigma(i)\sigma(\sqrt[4]{2})$. $\endgroup$ – Adam Hughes Apr 15 '15 at 7:42
  • $\begingroup$ @Adam Hughes If we want to get the subgroups of order 4 of the Galois group $G(K/ \mathbb Q)$ how can we do this? I mean if we want to get the subgroups of order $2$ we know that they are of the form $\{e,\sigma\}$ such that $\sigma^2=e$ where $e$ is the nuetral element. So what about the subgroups of order 4 for example? $\endgroup$ – Fareed Abi Farraj Jan 30 '19 at 21:19
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You should note that the Galois group is isomorphic to a subgroup of $S_4$, since any element of the Galois group will be determined by how it permutes the four roots of $x^4-2$. The order 8 subgroups of $S_4$ are isomorphic to the dihedral group with 8 elements.

If you want to know exactly what the eight maps are, there are four choices as to where $\sqrt[4]{2}$ is sent under some element of the Galois group. That tells you where the root $-\sqrt[4]{2}$ is sent as well, and then you have two choices as to where $i\sqrt[4]{2}$ is sent.

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