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This is a repost of my previous question, where I messed up everything. I make a explicit question here.

Suppose $\Bbb Q(\alpha, \beta)$ is a finite extension of $\Bbb Q$, where minipoly of $\alpha$ over $\Bbb Q(\beta)$ and over $\Bbb Q$ are different. How can I calculate the galois group by directly considering conjugate elements and write out all automorphisms that send $\alpha,\beta$ to conjugates where possible? i.e. I want to give an explicit construction of such galois groups.

For concrete examples you can consider $\Bbb Q(\sqrt2, \zeta_8)$

Note that I know $\Bbb Q(\sqrt2, \zeta_8)=\Bbb Q(\zeta_8)$, but I believe this is a coincidence and I can come up with another example for now. I want some sort of general solution to this kind of problems by directly considering conjugate elements, and $\mathrm{Gal}(\Bbb Q(\zeta_n)/\Bbb Q)\cong(\Bbb Z_n)^*$ doesn't provide such a solution to general cases. Is it possible?

Westlifer
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  • You speak about the GG of $\mathbb{Q}(\alpha,\beta)$. So you are assuming $\mathbb{Q}(\alpha,\beta)$ is a splitting field are you? Are you assuming the same about $\mathbb{Q}(\beta)$? – ancient mathematician Dec 26 '24 at 07:57
  • @ancient mathematician No, I didn't assume that. but if it is technically necessary, you can assume that, or even the extension is Galois, if needed? – Westlifer Dec 26 '24 at 09:56
  • A famous case is $\alpha=\sqrt[n]{a}$ and $\beta=\zeta_n$, in which case $\Bbb Q(\alpha,\beta)$ is the splitting field of $X^n-a$. Here the Galois group has been computed in general, see for example here. For $a=2$ and $n=8$ we have $\Bbb Q(\sqrt[8]{2},\zeta_8)$. – Dietrich Burde Dec 26 '24 at 10:28
  • You must be assuming $\mathbb{Q}(\alpha,\beta)$ is Galois [ie a splitting field since we are in char $0$] to pose this question. I don't see how it is answerable unless we know something about $\beta$ as well, eg its minpoly. My strategy would be to find a primitive element $\theta$ [I am not sure how to make it algorithmic, but some $\alpha+t\beta$, $t\in\mathbb{Z}$ will surely do]; calculate $m(X)$ the minpoly of $\theta$; use the standard algorithm to detemine the GG of $m(X)$ in terms of permutations of the conjugates of $\theta$; look to see what these perms do to $\alpha,\beta$. – ancient mathematician Dec 26 '24 at 13:29
  • Please do not answer in the comments, @ancientmathematician – Shaun Dec 26 '24 at 23:20
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    @Shaun. I don't think it's an answer, as I am still not clear about the status of $\beta$, are we given $\beta$ or are we to find all $\beta$ that split $m_{\alpha,\mathbb{Q}}$ while making $\mathbb{Q}(\alpha, \beta)$ Galois? I can't deal with that. – ancient mathematician Dec 27 '24 at 07:51

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